Fracture and Fatigue Emanating from Stress Concentrators

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

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Fracture and Fatigue Emanating from Stress Concentrators by

G. Pluvinage Université de Metz, Metz, France

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

1-4020-2612-9 1-4020-1609-3

©2004 Springer Science + Business Media, Inc. Print ©2003 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America

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CONTENTS Preface

vii

Chapter 1 Notch effects in fracture and fatigue 1.1 Notch effects in fracture 1.2 Notch effects in fatigue 1.3 Conclusion

1 1 7 15

Chapter 2 Stress distribution at notch tip 2.1 Introduction 2.2 Elastic stress distribution at notch tip 2.3 Stress distribution at notch tip for perfectly plastic material 2.4 Stress distribution for a elastic perfectly plastic material 2.5 Elastoplastic stress distribution for a strain hardening material 2.6 Conclusion

17 17 21 24 31 35 40

Chapter 3 Stress concentration factor 3.1 Definition of the stress concentration factor 3.2 Elastoplastic stress and strain concentration factor 3.3 Relationship between the elastic and elasto-plastic concentration stress and strain concentration factors and the elastic one 3.4 Evolution of elastic and elasto-plastic stress (or strain) concentration factor with net stress 3.5 Comparison of ] evolution with net stress 3.6 Conclusion

41 41 44

Chapter 4 Concept of notch stress intensity factor and stress criteria for fracture emanating from notches 4.1 Introduction 4.2 Concept of stress intensity factor 4.3 Concept of notch stress intensity factor 4.4 Global stress criterion for fracture emanating from notches 4.5 Local stress criterion for fracture emanating from notches 4.6 Notch sensitivity in mixed mode fracture 4.7 Conclusion Chapter 5 Energy criteria for fracture emanating from notches 5.1 Introduction 5.2 Influence of notch radius on the J integral 5.3 Influence of notch radius on the eta coefficients 5.4 Local energy criterion for fracture emanating from notches 5.5 Conclusion v

45 62 63 65 66 66 67 75 78 84 86 89 91 91 92 93 108 112

vi

CONTENTS

Chapter 6 Strain criteria for fracture emanating from notches 6.1 Introduction 6.2 Critical strain criterion for fracture emanating from notch 6.3 Strain distribution at the notch tip 6.4 Notch plastic zone 6.5 Conclusion

113 113 113 118 129 133

Chapter 7 The use of notch specimens to evaluate the ductile to brittle transition temperature; the Charpy impact test 7.1 History of the Charpy impact test 7.2 Stress distribution at notch tip of a Charpy specimen 7.3 Local stress fracture criterion for Charpy V notch specimens 7.4 Influence of notch geometry on brittle-ductile transition in Charpy tests 7.5 Instrumented Charpy impact test 7.6 Equivalence fracture toughness KIc and impact resistance KCV 7.7 Conclusion

135 135 137 140 141 144 149 153

Chapter 8 Notch effects in fatigue 8.1 Notch effects in fatigue and fatigue strength reduction factor 8.2 Relation between fatigue strength reduction factor and stress concentration factor 8.3 Volumetric approach 8.4 Influence of loading mode 8.5 Notch effects in low cycle fatigue 8.6 Conclusion

155 155 156

Chapter 9 Role of stress concentration on fatigue of welded joints 9.1 Introduction 9.2 Stress concentration factor in welding cords 9.3 Fatigue strength reduction factor 9.4 Standard methods for the design against fatigue of welded components 9.5 Innovative methods for the design against fatigue of welded joints 9.6 .Application of the effective stress concept to fatigue corrosion of welded joints 9.7 Conclusion

187 187 187 192 193 199

Chapter 10 Short fatigue grack growth emanating from notches 10.1 Short cracks emanating from smooth surface 10.2 Short cracks emanating from notches 10.3 The Role of the cyclic notch plastic zone 10.4 Stress intensity factor for short cracks and crack propagation 10.5 Conclusion

215 215 218 220 223 225

List of symbols

227

Index

231

164 170 179 184

209 213

Preface The vast majority of failures emanate from stress concentrators such as geometrical discontinuities. The role of stress concentration was first highlighted by Inglis (1912) who gave a stress concentration factor for an elliptical defect and later by Neuber (1936). In 1901 Charpy discussed the role of notch acuity on fracture energy whilst Schnadt indicated the necessity to use specimens with as high notch acuity as possible. Irwin developed the idea that defects should be considered as equivalent to cracks. Describing the stress distribution at the crack tip and introducing the concept of stress intensity factor. However, a crack is a mathematical cut of a plane and this leads to an infinite acuity and a stress singularity. Stress singularity cannot exist in reality because of stress relaxation which occurs by crack blunting, plasticity, damage, and finally fracture. However the concept of stress intensity factor can be helpful because fracture requires a fracture process volume in which the fracture stress acts as an average critical stress. Definition of this physical volume is difficult, however; this problem can be overcome by considering that the product of stress and the square root of distance is constant and proportional to the critical stress intensity factor. With the progress these has been in computing, it is now possible to compute the real stress distribution at a notch tip. This distribution is not simple, but in principle power dependence with distance remains and look like a pseudo-singularity. This distribution is governed by the notch stress intensity factor which is the basis of Notch Fracture Mechanics for which a crack is a simple case of a notch with a notch radius and notch angle equal to zero. Notch Fracture Mechanics is associated with the volumetric method which postulates that fracture requires a physical volume. In this volume acts an average fracture parameter in term of stress, strain or strain energy density. Since fatigue also needs a physical process volume; Notch Fracture Mechanics can easily be extended to fatigue emanating from a stress concentration. These different aspects are described in this book which summarise different researches studied carried out in the Laboratoire de Fiabilité Mécanique de l’Université de Metz. These research studies have been conducted in cooperation with numerous European institutions in the frame of an ’Open European Institute on Fatigue and Fracture’. The author would like to thanks all PhD students and foreign colleagues who have contributed to the development of Notch Fracture Mechanics. Thanks also to Bob Akkid who has read the final manuscript.

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CHAPTER 1 NOTCH EFFECTS IN FRACTURE AND FATIGUE

__________________________________________________________________ 1.1 Notch effects in fracture 1.1.1. DEFINITION OF NOTCH EFFECT IN FRACTURE Notch effect results in the modification of the stress distribution owed to the presence of a notch which changes the force flux (Figure 1.1). Near the notch tip the lines of force are relatively close together and this leads to a concentration of the local stress which is at a maximum at the notch tip.

Figure 1.1: Definition of maximum, gross and net stresses ; deviation of force lines due to the presence of a notch.

The local stress distribution exhibits a more severe gradient than the gross stress Vg shown as a uniform distribution in figure 1.1. The introduction of a notch in a component is more detrimental than the consequence of the net section reduction and leads to the following inequality: (1.1) Vmax > VN > Vg. 1

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Notch effects in fracture can be represented graphically, e.g., the critical gross stress versus a non-dimensional defect size. This graph is commonly known as Feddersen’s diagram [1.1]. This diagram is shown in figure 1.2 for the case of a simple plate loaded to have a uniform stress and having a central notch of length 2 a. The load bearing capacity of the ligament area is equal to the ultimate strength of the material. This leads to a linear decrease of the critical gross stress according to: (1.2) V cg Rm. 1  a W





(a notch length, Rm ultimate strength and W width). When a notch is present, the critical gross stress decreases with notch depth according to a non-linear relationship exhibiting a value less than that obtained from equation (1) with the exception of very small and large defects. The difference between experimental values of critical gross strain and theoretical values gets from equation (1) characterises the so called notch effect. The notch influence is strongly related not only to the dimensions of the notch but also to other geometrical parameters such as notch radius U and notch angle <. Values of these two parameters allow the following classification: crack :U = 0 and < = 0; infinite sharp notch:U = 0 and <  0; simple notch:U  0 and <  0. 1

Relative critical gross stress (Vgc/Rm) Plastic collapse VNc = Rm

Brittle fracture emanating from blunt notch A’

Increasing notch effect A

Brittle fracture emanating crack

1 Relative notch depth

Figure 1.2: Feddersen’s diagram exhibiting different fracture behaviour (AA’ is the notch effect).

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

3

A crack is the most dangerous form of notch. For brittle fracture, Linear Fracture Mechanics (LFM) indicates that the product of the square root of the notch size and the critical gross stress V cg equals to a constant [1.2]

V cg a

C1 ,

(1.3a)

From this diagram several types of behaviour can be described: x Plastic collapse (critical net stress equal to ultimate strength) Griffith criterion V cg a C 1 , x Griffith type criterion for a notch (product critical gross stress and notch depth at some power equal to constant). The severity of a simple notch is less than that of a crack and relation (1.2) is modified according to: (1.3b) V cg a E C 2 , x

where E is a constant. Quantitatively the notch effect (NE) can be defined as the relative difference between the simple consequence of cross section reduction and the real presence of a defect in term of the gross stress. If we notice that the critical gross stress after cross section reduction is equal to V cg , this definition can be written as : V cg *  V cg (1.4) NE V cg * with V c Rm.1  b / W , b is the ligament size below the notch. NE varies from 0 g* (no notch effect) to 0.2 ~0.3 in practice. 1.1.2 ELASTIC STRESS CONCENTRATION FACTOR We have seen that the presence of a notch in a loaded component leads to a maximum stress at the notch tip and a severe stress gradient. A typical example of such a distribution is given in figure 1.3. This figure relates to a round notched bar subject to a bending moment. The material behaviour is assumed to remain elastic. We can see that the stress distribution exhibits a maximum stress Vmax which is related to the gross stress Vg by the following relationship: (1.5) V max k t V g , where kt is the elastic stress concentration factor. The concept of a stress concentration factor is mainly associated with Neuber’s work [1.3]. According to some authors the elastic stress concentration factor is related to the net stressVN: (1.6) V max k t V N .

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Stress normal to notch plane Vyy (MPa) 400

300

E 360 STEEL 3PB

Vmax

200

Net stress level 100

3 0

0

2

1

3 Distance x (mm)

Figure 1.3: Elastic stress distribution at notch tip of a 3PB specimen (E 360 steel).

These two definitions sometimes lead to confusion as it is necessary to know whether the gross or net stress is being used. Vmax is most commonly determined using equation 1.5. A circular notch in a shaft for example has an elastic stress concentration kt = 2.6 which is calculated according to the Roark’s formula [1.4] k t f a w, a U , U is notch radius and D shaft diameter.

(1.7)

1.1.3 ELASTOPLASTIC STRESS CONCENTRATION FACTOR Owing to the high value of the stress concentration the maximum local stress is generally higher than the yield stress. Consequently a plastic relaxation occurs at the notch tip. In this case we can define the elastoplastic stress and strain concentration factors kV and kH according to the following formulae: kV

V max ;kH VN

H max HN

(1.8)

Vmax and Hmax are, respectively, the maximum stress and strain, VN and HN the net stress and strain. Several relationships exist between the elastic stress concentration factor kt

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

5

and the elastoplastic stress and strain concentration factor kV and kH; these are presented in chapter 3. Elastoplastic stress and strain concentration Factor kV and kH

STEEL E 550 CT SPECIMEN 3,5

kH

2,5 kV 0 0

150 300 Maximum stress Vmax

Figure 1.4: Evolution of the elastic and elastoplastic stress concentration factor and the elastoplastic strain concentration factor versus the maximum stress.

Values of the elastic stress concentration factor and elastoplastic stress and strain concentration factor have been obtained using finite element analyses for a CT sample with a notch depth 40 mm and of notch radius of 1 mm as presented in figure 1.4. The material is a steel of 550 MPa yield stress and 565 MPa ultimate strength. We can see that if maximum stress is less than the yield stress we have the following relationship: (1.9) k t k V k H for V max  Re If the maximum stress exceeds the yield stress, we find: k H t k t t k V for V max ! Re

(1.10)

1.1.4 FRACTURE CRITERION FOR FRACTURE EMANATING FROM NOTCH A simple criterion for fracture emanating from notches can be expressed based on the fact that fracture occurs when the maximum stress reaches the fracture stress Vf. at one point (generally the notch tip) Vmax = Vf. This approach is called the’ Hot spot approach’.

(1.11)

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Critical gross stress (MPa) 800

E550 STEEL SENT specimen c

V g = Rm . 1- a

600

W

400

200

Vmax = Vf 0 0

0.4

0.8 Non dimensional crack length (a/W)

Figure 1.5 : Comparison of the fracture criteria critical net stress or maximal stress equal to fracture stress and hot spot approach with experimental data ; SENT specimens made in E550 steel.

In figure 1.5 we have presented this fracture criterion and compared it with the fracture criterion critical net stress equal to fracture stress. Also are presented experimental results obtained on SENT specimens made in E550 steel with various ratios of notch depth a and width W. The fracture stress has been taken to be equal to the ultimate stress. We can see that the hot spot approach leads to a critical gross stress increasing with the ratio a/W. This is a consequence of the evolution of the elastic stress concentration factor with a/W. If we take into account the loss of constraint due to reducing ligament size, the evolution of kt with a/W is in opposite direction that the experimental results. This is the reason that the hot spot approach is inadequate for explaining fractures emanating from notches. A better approach is obtained by assuming that the fracture process needs a physical volume [1.5]. This assumption is supported by fracture resistance is affected by loading mode, structure geometry and scale effect. For this reason, it is necessary to take into account the stress value and the stress gradient in the neighbourhood at any point in the fracture process volume. This volume is assumed to be quasi-cylindrical by analogy with the notch plastic zone which has a similar shape. The diameter of this cylinder is called the effective distance. In order to take into account the neighbouring effect on the stress state in the fracture process volume, the stress at any point inside the process zone is weighted in order to take into account the distance from the notch tip and the relative stress gradient. Fracture stress can be

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

7

estimated by some average value of the weighted stresses. This is the so called ‘volumetric approach’ which will be described in chapter 4.

1.2 Notch effects in fatigue 1.2.1 EXPERIMENTAL EVIDENCE OF NOTCH EFFECTS IN FATIGUE Fatigue crack initiation is sensitive to notch effects as can be seen in figure 1.6 where the smooth specimen Wöhler’s curve is compared to a notched specimen. Stress range (MPa) 1000

E 360 steel

100

smooth notched 10 103

104

105

106

107

Number of fatigue cycles Figure 1.6: Notch effect in fatigue (E 360 steel; smooth and notched round bars subjected to bending moment).

For this, fatigue tests have been performed on a round notched bar subjected to a bending moment. The material is a steel E 360 (French standard) with a yield stress Re = 355 MPa and an ultimate strength Rm = 522 MPa. For the same number of fatigue cycles, the stress range is reduced by an important factor which characterises the fatigue notch effect. The smooth specimen Wöhler curve can be considered as the reference fatigue resistance curve for given conditions of specimen size, shape and loading state for the same material. It is represented generally by the Basquin’s law: 'V

V ' f  N R

b'

,

(1.12)

where 'V is the stress range,V’f the fatigue resistance, b’ Basquin’s exponent and Nr the number of cycle to fatigue failure.

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1.2.2 ROLE OF THE LOCAL MAXIMUM STRESS RANGE IN FATIGUE PROCESS Associated with the previous results, computation of the local maximum stress range by finite elements has been carried out. The local maximum stress versus the number cycles to failure has been plotted and compared with the fatigue reference curve obtained from smooth specimen. It can be seen that the curve 'Vmax,e f (NR) is above the fatigue resistance curve determined from smooth specimens (figure 1.7). In order to ensure the coincidence of these two curves it is necessary to reduce the value of the maximum stress to an effective stress value 'Vef. This can be done by two methods: the hot spot approach, and the volumetric method.

Maximum stress range (MPa) 1000 E 360 STEEL 500

smooth notched 10

0

103

104

105

106

107

Number of fatigue cycles

Figure 1.7: Maximum stress range versus number of cycles to failure (E 360 steel; notched round bar subjected to bending moment), comparison fatigue reference curve obtained from smooth specimens.

In the hot spot approach, we assume that the stress range at the initiation point plays a major role in the fatigue process but the effective stress range is less than the maximum stress range. We can see than if the stress distribution is elastoplastic, which is generally the case due to high stress concentration, the maximum stress range is a little distance behind the notch tip.In the volumetric method, we assume that the fatigue process needs a physical volume and the effective stress range is some average value of the stress range distribution in this volume.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

Stress normal to notch plane

9

Vyy

Vmax

Vef

A

Distance x Xef

Figure 1.8.a: Definition of effective stress according to Peterson [1.6].

Stress normal to notch plane Vyy

Vmax

Vef

B

Xef

Distance x

Figure 1.8.b: Definition of effective stress according Hardrath et al [1.7].

1.2.3 HOT SPOT APPROACH In order to take into account a maximum stress range less than the maximum stress, several ancient methods have been proposed in the frame of the hot spot approach.

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Peterson [1.6] proposed taking into account the stress range value on stress distribution at a distance Xef called the effective distance (Figure 1.8.a). Hardrath et al [1.7] proposed an average stress range over an effective distance (Figure 1.8.b). Neuber [1.3] has proposed to increase fictitiously the notch radius which becomes equal to Xef. Consequently the maximum stress range decreases (Figure1.8.c). For these authors the effective distance is an empirical constant with no physical meaning which depends generally on material and mechanical properties Stress normal to notch plane Vyy

Vmax Distribution for notch radius U

Vef

Distribution for notch radius U +Xef Xef

Distance x

Figure 1.8.c: definition of effective stress according to Neuber [1.3].

1.2.4 VOLUMETRIC METHOD. Mechanisms of crack initiation have been described widely. They consist of intrusion and extrusion mechanisms in pure ductile metals or dislocation pile ups on inclusions, decohesion of the matrix and finally crack initiation. Two major elements indicate that the fatigue mechanism requires a physical volume to take place: ífatigue tests are generally affected by a large scatter. A description of the scatter by the Weibull approach is well documented. This leads to the assumption that the probability of fatigue initiation is proportional to the process volume where the probability to find an initiation site (e.g, non-metallic inclusion) is assumed to be uniform; ífatigue resistance is influenced by the size of the specimen and the relative stress gradient (i.e, the stress gradient divided by the stress value) which are dimensional parameters. It has been seen that fatigue resistance decreases with the volume process zone containing stresses distributed within the range [100~90] % of maximum stress.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

11

These arguments lead to the conclusion that the fatigue mechanism requires a physical volume in which to take place. Within this volume the average stress is high enough to promote fatigue initiation, the relative stress gradient is not high enough in order for all points inside this volume, the ‘effective volume’ to be sufficiently stressed. The role of the relative stress gradient on the fatigue process was previously discussed by Buch [1.8]and Brandt [1.9]. The volumetric approach overcomes the disadvantage of the ‘hot spot approach‘. In addition, no empirical relationships between the elastic and the fatigue stress concentration factor exist. The cyclic stress strain curve of the material, the intrinsic fatigue resistance curve and the computation of the notch tip elastoplastic stress distribution are the elements necessary for a prediction of the high cycle fatigue life time. Logarithm of stress range 'Vyy

'Vmax

'Vef

Effective distance Xef

Logarithm of distance (log r) Figure 1.9: Definition of the effective stress according to the volumetric method [1.5].

This approach leads to a two parameter fatigue initiation criterion, the effective stress and the effective distance. The effective distance is associated with a particular point of the elastoplastic stress distribution presented in a bilogarithmic graph. The effective stress corresponds to an average value over this distance of the stress distribution weighted by the distance and the relative stress gradient (figure 1.9). This is presented in chapter 8. 1.2.5 FATIGUE STRENGTH REDUCTION FACTOR Because the elastic or the elastoplastic stress concentration factor cannot give directly the effective stress range in fatigue, another parameter was introduced: the fatigue

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strength reduction factor. Initially the fatigue strength reduction factor was defined as the ratio of the endurance limit measured on a smooth specimen and that of a notched specimen: kf

V D, s V D, n

,

(1.13)

where VD,s is the endurance limit measured on a smooth specimen ; VD,n is the endurance limit measured on notched specimen. The notion of the fatigue strength reduction factor has been widened to any life duration and can be considered as related to the effective fatigue stress: (1.14) 'V ef k f . 'V g , where 'Vef is the effective stress range, 'Vg is the gross stress range applied to a notched specimen leading to a life duration NR.

Fatigue strength reduction factor

E 360 steel round notched bars in bending

2.5

kt

1.5

Maximum stress plastic Maximum stress elastic

1

103

105 number of fatigue cycles

107

Figure 1.10: Evolution of the fatigue strength reduction factor with the number of fatigue cycles (E 360 steel, round notched bars in bending).

The fatigue strength reduction factor for a single edge notch tensile (SENT) specimen is a function of the notch radius as shown in figure1.11. It increases with the decreasing of notch radius for the same value of the applied stress range. This second definition leads to a fatigue stress concentration factor which varies with the number of cycles to failure through the applied stress range as can be seen on figure 1.10. For a low number of cycles, kf is near to unity; for a high number of cycles, it tends asymptotically to the value.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

13

Fatigue strength reduction factor kf

U = 0.1mm

9

U =0.2 mm U =0.3 mm 7

5 0.3

0.2

0.4

Non dimensional net stress (VN/Re) Figure 1.11: Evolution of the fatigue stress concentration factor kf versus net stress VN divided by yield stress Re (SENT specimen).

1.2.6 FATIGUE SENSITIVITY INDEX To express the difference between the fatigue strength reduction factor kf and the elastic stress concentration factor the fatigue sensitivity index is generally use, three definitions of this index have been proposed : the oldest [1.6] q1

k f 1 ; k t 1

(1.15) the most often used [1.4]

q2

kf kt

;

(1.16) one of the most recent [1.5]

kf

; (1.17) kV where kVis the elastoplastic stress concentration factor. q3

Numerous empirical relationships have been proposed for expressing the notch sensitivity index (determined for the endurance limit) versus the notch radius. These

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are presented in chapter 8. The accuracy of these formulas is not known and needs to be checked by numerous, costly and time consuming fatigue tests. Fatigue sensitivity index

1.0

0.6 Heywood Neuber Peterson

0

2

4 Notch radius (mm)

Figure 1.12: Evolution of fatigue sensitivity index as a function of the notch radius variation. Comparison of 3 relationships of the literature. CT specimen E360 steel.

The use of the notch sensitivity in order to describe the notch effect on high cycle fatigue presents the following disadvantages the empirical relationship between kf and kt are established only for the endurance limit, , they incorporate empirical constants which were determined many years ago; the steel quality and particularly the inclusion content was different to present steels, also the validity of the constants is questionable;, , the difference between the relationships is relatively high and the accuracy of these formulas is unknown; , the influence of the net stress and stress gradient is not taken into account. The evolution of the fatigue sensitivity index with the notch radius has been obtained using three relationships proposed in literature: those from Neuber [1.3] (relationship 1.18), Peterson [1.6] (relationship 1.19), and Heywood [1.10] (relationship 1.20) and plotted in figure 1.12. q1 q1

>

1

>

U

1   U U '

U ' 1

@

1

@ ,

,

(1.18) (1.19)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

q

2

>1   2U U ' @1 ,

15

(1.20)

U’ is a constant which differs for each author (0.040 for Neuber, 0.105 for Peterson, and 0.015 for Heywood for steel E360). The notch sensitivity index trends asymptotically to 1 when the notch radius increases but this evolution differs from the literature models.

1.3 Conclusion Notch effects present some similar approaches in fatigue and fracture and are characterised in fracture by the critical gross stress being less than resulting only in a reduction of the cross section and in fatigue by the fact that the Wöhler curve obtained with a notched specimen being below that obtained with a smooth specimen. In any case, the value of maximum stress or maximum stress range at the notch tip is able to describe this effect; experimental results show that the effective critical stress or effective stress range has a lower value than the maximum stress or stress range. To model this phenomenon two major approaches are possible: the hot spot approach, the volumetric approach. The hot spot approach considers that the stress or the stress range value, for some particular point on the stress distribution, is equal to the effective stress or stress range. The volumetric approach assumes that fatigue or fracture needs a physical process volume. Inside this volume at any point stress plays a role which depends upon its distance to the notch tip and the stress gradient. The hot spot approach is historically more traditional but cannot explain loading mode and scale effects in fatigue and fracture. The effective stress or stress range being less than the maximum stress or stress range cannot be explained by plastic or damage relaxation but that a part of the stress distribution is acting on the process with an average value less than the maximum one. For the particular case of a notch as a crack the volumetric approach seems more appropriate owing to the assumption of a stress singularity in which the maximum stress or stress range is infinite. There is a continuous approach between notch fracture mechanics and classical fracture mechanics.

16

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REFERENCES 1.1 Feddersen.C.E. (1970). ‘Evaluation and prediction of residual strength of center cracked tension panels’. ASTM STP 486, p 50. 1.2 Griffith.A.A. (1920). Phil Trans. Roy. Soc. London, A.221, , pp163~198. 1.3 Neuber. H. (1958). ‘Kerbspannungslehre’ Springer: Berlin, "(in german). 1.4 Roark. R.J.(1989). ‘Roark’s Formulas for stress and strain’. 6th Edition MacGraw-Hill International Editions, 1.5 Boukharouba. T, Tamine .T, Nui.L. Chehimi.C ,Pluvinage G.(1995).’The use of notch stress intensity factor as a fatigue crack initiation parameter’. Engng. Fract. Mech, Vol N°. 3, pp 503~512. 1.6 . Peterson.R.E(1959). ‘Notch Sensitivity’ . Metal Fatigue, (Edited by G.Sines & J.L. Waisman), MacGraw Hill, New-York, 1.7 Hardrath. H.F. and Ohman .L. (1953).’ A study of elastic and plastic stress concentration factors due to notches and cracks on flat plates’. NACA-Report 1117, 1.8 Buch.A.(1974). ‘Analytical approach to size and notch size effects in fatigue of material specimens’. Materials Science and Engineering, Vol. 15, pp 75~85. 1.9 Brandt A. (1980). ‘ Calcul des pièces à la fatigue par la méthode du gradient ‘.Editeur CETIM. 1.10 Heywood.R.B.(1952). ‘ Designing by photo-elasticity’, Chapman and Hall, London, UK.

CHAPTER 2 STRESS DISTRIBUTION AT NOTCH TIP

________________________________________________________ 2.1 Introduction: stress distribution around an hole In order to have some idea of the stress distribution around a geometrical discontinuity, let us consider a bi-dimensional plane stress state of a circular hole in an infinite plate subjected to a bi-axial state of stress V1 and V2. We assume that the behaviour of the material is always elastic.

V1 B V2 A

Figure 2.1: Plate with a circular hole submitted to a biaxial state of stress V1 and V2.

We can write the equilibrium equations in polar co-ordinates: w V rr 1 w V rT V rr  V rT   0, wr r wT r 1 w V TT w V rT 2V rT   wr r wT r

17

0,

(2.1)

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where r and T are polar coordinates. The boundaries conditions are: the radial stress Vrr at the hole surface is zero owing this surface being a free boundary. For infinite r, the stress field is identical to that of a plate without a hole. case I : Equi-biaxial tension V1 = V2 = S  By symmetry, the solution is independent of Tand the shearing stress is zero due to isotropy. The hole radius is equal to unit (r =1). For infinite rVrr =VrT = S; the following distribution is proposed :  * B*   V TT C *  D ,   V rr A*  2 2 r r The first equilibrium equation leads to: B*  D* (2.3)  0  r3 where A*, B*, C* and D* are constants to determined. This condition is satisfied for any value of r value so B* = -D*. On the hole surfaceVrr = 0 and S + B* = 0. All the constants are known and we have: § § 1 · 1 ·   V rr S ¨¨1  ¸¸  V TT S ¨¨1  ¸¸    © r2 ¹ © r2 ¹ Case II : pure shear V1 = - V2 = S We have the following boundary conditions

r of ; T= 0

 Vrr

 VTT

+S

-S

-S +S r of ; T=S/ 2   WrT = 0 for T = O and T= S/2. It is necessary to have a distribution to the power (-4). The following form is proposed: § C * D * ·¸  cos 2T , V TT ¨¨  S  ¸ r2 r4 ¹ ©

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

B * ·¸ cos 2T ¸ r4 ¹ § F * G * ·¸  sin 2T . V rT ¨¨ E *  ¸ 2 r r4 ¹ © Equilibrium equations lead to: -

2A r

S-

C r

*

3

*

2



4B



r

D r

5

*

cos 2T 

  r

*

4

cos 2T 

-

*

E 

2F r

A*  r2

§ ¨S  ¨ ©

V rr

3

*



F

*

r

2

4G

*

r

19



5

G r

*

*

*

4

(2.5)

cos2T 

*

sin2T 

*

*

F G * 2   E + 2 4 r r r

sin2T



(2.6) These equations are equal to zero for any values of r and T. So we have: E*

+S

- A*

+ 2F*

- 3B*

+ 3G*

- 2C* D*

In addition Vrr is equal to zero on the hole surface S +A* + B* = 0 The shear stress is zero for infinite r, V rr  V TT sin 2T . W rT  2 7 equations for 7 unknowns leads to :

= 0, - C*

= 0,

= 0, = 0,

+ G*

*

B S A S C D +  +   cos2T 5 3 r r 3 5 r r r r

= 0,

(2.7)

(2.8)



20

G PLUVINAGE

§ 4 3 · S ¨¨1   ¸¸ cos 2T , © r2 r4 ¹ § 3 · V TT  S ¨¨1  ¸¸ cos 2T , 4 © r ¹ § 2 3 · V rT  S ¨¨1   ¸¸ sin 2T . © r2 r4 ¹

V rr

(2.9)

General case Any bi-dimensional stress state V1 and V2 can be considered as the sum of an isotropic state (V1 +V2)/2 and a pure shearing state (V1 -V2)/2. By superposition we get: ªV  V § 1 ·º ª V  V § 4 3 ·º V rr « 1 2 .¨¨1  ¸¸»  « 1 2 .¨¨1   ¸¸». cos 2T , «¬ 2 © r 2 r 4 ¹»¼ © r 2 ¹»¼ «¬ 2

V rr

ªV 1  V 2 § 1 ·º ª V  V 2 § 3 ·º .¨¨1  ¸¸»  « 1 .¨¨1  ¸¸». cos 2T , « «¬ 2 © r 2 ¹»¼ «¬ 2 © r 4 ¹»¼

W rT

ªV  V 2 § 2 3 ·º « 1  ¸¸». sin 2T . .¨¨1  «¬ 2 © r 2 r 4 ¹»¼

(2.10)

At the hole surface (r =1) we have Vrr = 0 and VTT = (V1 +V2) - 2(V1 -V2) cos2T. The circumferential stress has a maximum of 3V1 -V2 and 3V2 -V1. Unaxial case From the previous solution with V2 = 0 we obtain: º 1 · § 4 3 · V 1 ª§¨ .«¨1  ¸¸  ¨¨1   ¸¸. cos 2T » , V rr 2 «¬© r 2 ¹ © r 2 r 4 ¹ »¼ º 1 · § 3 · V 1 ª§¨ .«¨1  ¸¸  ¨¨1  ¸¸. cos 2T » , V rr 2 ¬«© r 2 ¹ © r 4 ¹ ¼»

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

ªV § 2 3 ·º  « 1 .¨¨1   ¸¸». sin 2T . 2 r 4 ¹» ¼ ¬« 2 © r

W rT

21

(2.11)

In the following the stress distribution is described for the following material behaviour: elastic, rigid plastic, elastic perfectly plastic, and strain hardening. 2.2 Elastic stress distribution at notch tip

In order to have some idea of the influence of the notch radius on the elastoplastic stress distribution at the notch tip, let us consider a three point bend specimen subjected to a load P (Fig 2.2). The applied net stress VN on the ligament below the notch is given b :

3PL

VN

2.B. W  a

(2.12)

2

where L is the span, W the width and a the notch depth. Assessment of the influence of the notch radius on the elastic stress distribution at the notch tip can be seen in Figure 2.3, where the stress normal to the notch plane VYY divided by the net stress is plotted against the distance r divided by notch depth a for different ratios of U/a where U is the notch radius. We can see that this stress distribution exhibits a maximum stress at the notch tip which increases strongly with decreasing of notch radius. P L

W

U

<

P/2

P/2

Figure 2.2: Geometry of a three points bending specimen.

22

G PLUVINAGE

Ratio stress normal to notch plane to net stress (Vyy/VN) P

100

r

80

U/a = 6.10-4 U/a = 5.10-3 U/a = 13.10-3 U/a = 15.10-3 Ua = 20.10-3 U/a = 30.10-3 Ua = 40.10-3 U/a = 56.10-3

Vyy

a \

60

P/2

P/2

40 20 0

0

0.01

0.02

0.03

0.04

0.05

Non dimensional distance r/a

Figure 2.3: Evolution of the stress normal to the notch plane VYY divided by the net stress with the distance r divided by notch depth a for different ratios of U/a computed by Finite Element Method.

This is the stress concentration produced by the presence of the notch. The stress gradient at the notch tip also increases strongly with the decreasing of the notch radius. from a square root dependence. 2.2.1 ANALYTICAL FORMULAE FOR ELASTIC STRESS DISTRIBUTION AT THE NOTCH TIP Different analytical formulae summarised in table 2.1.

for elastic stress distribution at notch tip are

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

Table 2.1 Analytical formulae for stress distribution at notch tip.

23

24

G PLUVINAGE

Figure 2.4: Comparison of different analytical formulae for the elastic stress distribution at a notch tip. Comparison with a distribution obtained from Finite Element Method; (Case of a CT sample).

2.2.2 COMPARISON OF ANALYTICAL FORMULAE AND CONCLUSION The validity of some of these analytical formulae has been compared with results obtained using a Finite Element Method on a CT specimen loaded with a net stress of 485 MPa. The notch radius is taken as 0.75 mm.. We can see that the difference between all these stress distributions is small near the notch tip and a difference appears remote from this point. In our case, Chen’s solution gives the nearest results to the finite element method (figure 2.4). 2.3 Stress distribution at notch tip for perfectly plastic material

If we consider a rigid and perfectly plastic material, any point we consider in the direction of the maximum shear stress and draw a line which is always normal to the maximum shear direction. In a body which deforms plastically such a line represents the material flow. These lines are known as slip lines. Any lines which deviate from 45 °in the positive (anti-clockwise) sense of the maximum principal stress are called D lines. The lines orthogonal to D lines are called E lines figure 2.5).. Hencky’s theorem gives the properties of these lines:

Vm 2k

Vm 2k

T

T

cst

cst K

[

along D lines,

(2.29)

along E lines,

(2.30)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

25

V

1

E line 45°

D line 45° V2

Figure 2.5: Schematic representation of the D andE slip lines.

where T is the deviation angle of the line (from the reference direction), Vm the hydrostatic pressure and k the shearing stress. Intuitively slip line fields are drawn from experimental evidence of the plastic deformation of the material and therefore the solution is non unique. 2.3.1 PLATE WITH A CENTRAL HOLE In this case, the slip line field can be represented by the following fields: í,logarithmic slip line field (the logarithmic line has the property of always intersecting a radial line from the origin at an angle of 45° ) ; ía constant stress field with linear slip lines (In this case the angle T is zero and the stresses are constant in the field) ; íany combination of the two previous slip line field. *Stresses in the logarithmic slip lines field Radial stress in the logarithmic slip lines field V rr

§ r· 2k ln¨¨1  ¸¸ . © U¹

(2.31)

Circumferential stress in the logarithmic slip lines field VTT = Vrr+2k.

(2.32)

26

G PLUVINAGE

Figure 2.6: Slip lines stress field for a plate with a circular hole. a) logarithmic slip line stress field, b) constant slip line stress field, c) mixed slip lines stress field.

Stresses in the homogeneous slip lines field V xx = W xy  V yy = 2k .

(2.33)

Value of the limiting load r1 w 2 ³ V TT  2 r1 ³ V yydx, (2.34) a r1 ª r1 r º P 4k «W  a  ³ ln dr » (2.35) a » « a ¬ ¼ rl is the width of the logarithmic slip line field. We can see that the limiting load is maximum for case b and equal to P = 4K.(W-a) and minimum for casea ( logarithmic slip line stress field). P

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

2.3.2 DOUBLE EDGE NOTCH PLATE (NOTCH ANGLE< )

P

C

B A

D

b

P

Figure 2.7: Slip line stress field for a double edge notch plate (notch angle< ).

In the field OAB

V m k ,T

In the field ODC

T

S 4 , K

 3S 4 , K

V m k 1  S

1 2 S 4 ,

V m 2k  3S 4 1 2  3S 4 , ,

V yy

k 2  S , V xx

kS

.

(2.36)

The limit load is equal to (with P*L = 2kb):

PL P*L

§ S· ¨1  ¸ . © 2¹

(2.37)

27

28

G PLUVINAGE

2.3.3 DOUBLE EDGE NOTCH PLATE (NOTCH ANGLE
P

Zone B Zone A

Zone C <

<

D A B C

b

P

Figure 2.8: Slip line stress field for a double edge notch plate (notch angle
In this case the limit load is equal to: PL S \ = 1+  2  * PL

(2.38)

2.3.3 DOUBLE EDGE NOTCH PLATE (NOTCH ANGLE< AND B/2Ud) A particular case is obtained for the case where b/U is ” 3.81 = eS/2. This is the case of
FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

\

§ r · ¸¸ ln¨¨ © 2U ¹

(2.39)

P

2k (1 + Lnb/a) 2k

B

b

P

Figure 2.9: Slip lines stress field for a double edge notch plate; (notch angle< and b/2U ” 3.81).

This solution for the circumferential stress distribution is known as Hill’s solution [2.11]. By Integration we get the limit load: PL

PL P*L

b 2 2 ³ V TT dr , a b · § 2U · § ¸ . ¸. ln¨1  ¨1  b ¹ ¨© 2 U ¸¹ ©

(2.41)

(2.42)

29

30

G PLUVINAGE

2.3.5 NOTCH SPECIMEN SUBMITTED TO A BENDING MOMENT In this case the stress field is described in figure 2.10. The field BCB consists of a logarithmic slip line field of length b*. The field CAA is a constant stress field. In the logarithmic slip line field, the radial stress is given by: §r· (2.43) V rr 2k ln¨¨ ¸¸ . ©U¹ O B b*

M b

M C D lines

E lines

VTT

A

-2k

0’

A’

Figure 2.10: Slip lines stress field for notch specimen subjected to a bending moment.

The circumferential stress is given by: for U ” r ” U+ b*

V TT

ª § r ·º 2k «1  ln¨¨ ¸¸» , © U ¹¼ ¬

for  U+ b* ” r ” U+b

V TT

2 k ,

(2.44)

The limiting bending moment is equal to:

U  b* ³ V TT .rdr . U We define the parameters ] and ]1 by: ML

[1 1 Finally

(2.45)



b*

U

,[

1

b

U

.

(2.46)

31

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

ML M *L

2 2 2 º ª  « 9  [ 1  9  2 [ 1  7 9 2  8 [ 1». 9  [ 1 ¼ ¬

M *L

k b2 2











(2.47)

2.4 Stress distribution for an elastic perfectly plastic material

2.4.1 WILSHAW ET AL [2.12] ANALYSIS Wilshaw et al has combined Hill’s solution with the elastic stress distribution for a notched beam subjected to a bending moment M. The stress distribution along the ligament is divided into 4 parts (Figure 2.11): ízone 1 for 0 ” r ”. Ry: The circumferential stress is given by the Hill’s solution ª § x ·º (2.48) ı yy Re «1  ln¨¨1  ¸¸» , ȡ ¹¼» © ¬« The maximum stress is reached when x = Ry ª § R y ·º ¸» , Re «1  ln¨1  ¨ U ¸¹»¼ «¬ © and Ry depends of the value of < angle according to :

V yy

max

ª § R y ·º ¸» «ln¨1  ¨ U ¸¹»¼ «¬ ©

ª S <º «1  2  2 » ; ¼ ¬

(2.49)

(2.50)

í zone 2 for RE” r ”. Ry : the stress is constant and equal to the maximum stress : ª S <º Re «1   » ; ¬ 2 2¼ ízone 3 for RE” r ” RN: the elastic stress distribution is used :

V max yy

V YY V l ( x)

U . U  4r

(2.51)

(2.52)

32

G PLUVINAGE Vyy ( r )

U . U 4r

k V VN ( x )

(2.53)

where VN is the net stress and Rn the distance of the neutral axis.

0

Vyy 1

Ry 2

RN M

3 M

RE Stress distribution along ligament 4 r

Figure 2.11: Stress distribution along the ligament of a notched beam under bending. Wilshaw et al’s solution .

The elastoplastic stress concentration factor kVis given by   ª § R E ·º ¸» , k V «1 Ln¨¨1  U ¸¹¼» © ¬«

(2.54)

íZone 4 : for R1” r ”.b: the stresses are compressive and the distribution is given for the solution of an the elastic beam under bending.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

33

2.4.2 BHATTACHARYA ET AL’S ANALYSIS Bhattacharya et al’s [2.9] proposed a solution for the stress distribution along the ligament of a notched beam under bending using the superposition of 4 or 5 stress distributions (VI ,VII ,VIII ,VIV ,VV). This solution, for an elastic perfectly plastic material, is similar to those proposed for an elastic material. x

For small scale yielding (Figure 2.12) ı yy ı I  ı II  ı II  ı IV ,

(2.55)

VI , VII et VIII are tensile stresses and VIV a compressive stress distribution. These stress distributions act in different zones. í stress VI for 0 ” r ” Ry

VI

ª § x ·º Re «1  ln¨¨1  ¸¸» . U © ¹¼ ¬

(2.56)

Stress normal to notch plane Vyy Re VN

VI

Ry

VII

VIII

Distance x

VIV RE l (W-a)/2

VV W-a)/2

Figure 2.12: Stress distribution along the ligament of a notched beam under bending.

34

G PLUVINAGE

ístress VII for Ry ” r ” l k t .V N .

V II

U . U  4r

(2.57)

The l distance is given by

U





. k2 1 , 4 t where kt is the elastic stress concentration factor. l

(2.58)

ístress VIII for 0 ” r ” (W-a)/2

V III

M § W  a · .¨ r¸ . I © 2 ¹

(2.59)

M I

(2.60)

ístress VIV for (W-a)/2 ” r ” (W-a)

V

IV

W a· § ¸ ¨r  2 ¹ ©

Stress normal to notch plane Vyy Re VN

VI

Ry

VII

VIII

Distance x

VIV RE l (W-a)/2

VV W-a)/2

Figure 2.13: Stress distribution along the ligament of a notched beam under bending. Bahattacharya et al’s

x

For general yielding (see Figure 2.13)

Solution for general yielding [2.9]. 

Vyy = VI + VII +VIII +VIV +VV,

(2.61)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

35

VI ; VII ; VIII and VIV are tensile stresses and VV a compressive stress distribution. í stress VI for 0 ” r ” RE

VI

ª § x ·º Re «1  ln¨¨1  ¸¸» , U © ¹¼ ¬

(2.62)

V II

ª S <º Re «1   » , ¬ 2 2¼

(2.63)

í stress VII for RE ” r ” Ry -

í stress VIII for Ry ” r ” l

V III

k t .V N .

U , U  4r

(2.64)

- stress VIV for 0 ” r ” (W-a)/2 M § W a· ¸, ¨r  I © 2 ¹ - stress VV for (W-a)/2 ” r ” (W-a) M § W a· ¨r  ¸. VV I © 2 ¹

V IV

(2.65)

(2.66)

2.5 Elastoplastic stress distribution for a strain hardening material

2.5.1 ANALYTICAL FORMULAS 2.5.1.1 Tetelman and MacEvilly [2.13] have modified Hill’s formula in order to take account of strain hardening. For this reason they use the flow stress R c instead of the yield stress in the formula: ª § x ·º 2 Rc «1  ln¨¨1  ¸¸» . (2.67) V yy U 3 ¬ © ¹¼ 2.5.1.2 Xu [2.14] has proposed a relationship to describe the stress distribution at the notch tip for a notched beam subjected to pure bending in plane strain conditions. The material is assumed to have a behaviour following the RambergíOsgood law:

36

G PLUVINAGE

H H0



V V  D. V0 V0

1

n

,

(2.68)

V0 is the reference stress,H0 is the elastic reference strain, D a constant and n the strain hardening exponent. The effective strain Hef in plane strain conditions is: 2 2 2 (2.69) .  H x  H Y  H x 2  H Y 2  1.5 J xy , 3 Hx and Hy are the strains in the x and y directions and Jxy the shearing strain. The effective stress Vef is given by :

H ef

V eq

1 2

.

 V 1  V 2

2

2 2   V 2  V 3   V 1  V 3 ,

(2.70)

where V1, V2 et V3 are the principal stresses V1 > V2 > V3. The effective plastic strain Hpef is obtained from: 1 H pef H ef V ef V ef n  . (2.71) D. H0 H0 V0 V0



Maximum principal stress/ reference stress

Maximum stress ahead the notch tip Notch tip

Non dimensional distance x/U

Figure 2.14: Stress distribution at notch tip according to Xu et Al [2.14].

The principal stress distribution in the plastic zone is given by : § § x ·ª x ·º V1 Q r exp¨¨  I .C.n. ¸¸.«1  ln¨¨1  ¸¸» , (2.72) U ¹¬ V0 © © U ¹¼ C is a constant which depend of the strain hardening exponent.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

Qr is defined from:



n

H pef 3 DH0

2

Qr

,

37

(2.73)

I Hpef, the plastic effective strain at the notch root depend on the strain hardening exponent. The stress distribution is given in Figure 2.14. We can see that the maximum stress is not at the notch tip in this case. The Hill formula can be replaced by the Lin et Al formula [2.15]. U§ V ef · º 2 ª ¸dr » , .«V ef  ³ ¨ V1 ¨rU¸ » 3 «¬ ¹ ¼ 0© which leads to another expression n

V1 V ef







H pf

r

 exp C.n.

D H ef

(2.74)

r U

+

exp C.n.

r U

.

dr rU

.

(2.75)

0

This formula overestimates the stress distribution compared to the Finite Element method. For this reason a correction factor I is introduced for 0  I  1 and ] = r/U H n I .C.[ , (2.76) §H ln¨¨ p ef © Hp

with H n

· ¸¸ and I = 0,85 . ¹

The final expression is: n

V1 V ef

 



H pf D H ef

 exp C.n.

r U

. 1 + Ln 1 +

r U

0 < r < RE .

(2.77)

2.5.2 FINITE ELEMENT METHOD [2.16] The stress distribution at the notch tip for an elastoplastic material has been computed using Finite Element method. A CT specimen with four different notch radii (U = 0.1 ; 0.5 ; 0.75 and 1 mm) has been studied. The dimensions of the specimen are: thickness B = 15 mm ; width W = 80 mm ; Height L = 96 mm ; ligament size b = W - a = 36 mm ; Notch angle \ = 40°. The material is a E 550 steel according to French standard. Mechanical properties are as follows :Yield stress( Re = 572 MPa) ; ultimate strength (Rm = 684 MPa).

38

G PLUVINAGE P

L/2

a

U

W

Figure 2.15 : CT specimen geometry (half section).

Ratio of the stress normal to notch plane to the net stress (Vyy/VN)

1.00

0.75

See figure 2.17

0.50

P/Pc% 10 20 30 40 50 60 70 80 90 100

0.25

0.0

0.1

0.2

0.3

0.4

0.5

Non dimensional distance r/a

Figure 2.16 : Elastoplastic stress distribution at the notch tip of a CT specimen.

We can see on figure 2.17 the influence of load level: í for the first step (10% of critical load), the stress distribution is elastic and has a dependence 1/¥x near the notch root . The maximum stress is at the notch tip. í for the other steps, plastic deformation appears and the maximum stress is not at the notch tip. The stress distribution changes with the load level.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

39

ífor low load values, the stress distribution can be divided into two parts : firstly the stress increases to a maximum stress and secondly decreases elasticity with a 1/¥x dependence. ífor higher load values : the stress distribution can be divided into three parts : first the stress increases to maximum stress, then decreases slowly and finally decreases elastically with a 1/¥x dependence. The notch radius has a strong influence on the maximum critical stress. The stress distribution computed by the Finite element method has been plotted as the non dimensional stress normal to the notch plane Vyy/VN versus the non dimensional distance x/b, VN is the net stress and “a” the notch length. For a large notch radius, the maximum stress for a critical load is slightly higher than the ultimate stress (1.13 Rm for U= 1 mm and 1.16 Rm for U = 0.75 mm) and situated at a small distance from the notch tip. For a small notch radius, the increase in the critical maximum stress is more pronounced (1.24 Rm for U= 0,5 mm and 1.64 Rm for U = 0.1 mm) and situated practically at the notch tip Ratio of the stress normal to notch plane to the net stress (Vyy/VN) 1.25

P/Pc% 20 30 40 50 60 70 80 90 100

CARBON STEEL CT SPECIMEN 1.00

0.75

0.50 0.00

0.05

0.10

0.15 Non dimensional distance r/a

Figure 2.17: Elastoplastic stress distribution at the notch tip of a CT specimen (enlargement of the previous graph).

0.20

40

G PLUVINAGE

2.6 CONCLUSION

The stress distribution is different for an elastic material from those of an elastoplastic one: the maximum stress is found at the notch tip in the first case and a little distance behind for the second case. Generally the dependence is more complicated than that of the 1/—x dependence.

REFERENCES 2.1 Thimoshenko. S,. Goodier. J .N., (1951). ,’Theory of elasticity.’,McGraw-Hill, New York, 2.2 Neuber.H., (1961). ‘Theory of stress concentration for shear-strained prismatic bodies with arbitrary nonlinear stress-strain law’. Journal of Applied Mechanics, 28, p 254, 2.3 Creager. M, Paris. P. C., (1967).’Elastic field equations for blunt cracks with reference to stress corrosion cracking’. International Journal of Fracture, Vol 3, pp 247~255, 2.4 Chen. C .C., Pan H .I., (1978).’Collection of papers on fracture of metals’.(C. Chen Ed ) Metallurgy Industry Press, Beejing, pp 197~21, 2.5 Usami. S., (1985). ‘Short crack fatigue properties and component life estimation’. Current Research on Fatigue Crack, Edited by Tanaka. T.; The Society of Materials Sciences, Kyoto, Japan. 2.6 Glinka. G., Newport. A., (1987). ‘Universal features of elastic notch tip stress field.’ International Journal of Fatigue, Vol. 9, N° 3, , July, pp 143~150. 2.7 Kujawski. D., (1991).’Estimation of stress intensity factor for small cracks at notches’. Fatigue Fracture Materials and Structures, Vol. 14, N° 14, pp 953~965. 2.8 Xu. Kewein, He. Jiawen, (1992). ‘Prediction on notched fatigue limits for crack initiation and propagation.’. Engineering Fracture Mechanics, Vol. 41, N° 3, pp 504~410. 2.9 Bhattacharya. S., Kumar. A .N., (1995). ‘Rotational factor using bending moment approach under elastoplastic ; situation I : Notch 3PB Geometry’. Engineering Fracture Mechanics, Vol. 50, N° 4, pp 495~505. 2.10 Xu. R.X., Thompson. J.C., Topper. T.H. (1995).’Practical stress expressions for stress concentration regions’. Fatigue and Fracture of Engineering Materials and Structures, Vol. 18, N° 718, pp 885~895. 2.11 Hill. R., (1959).’The mathematical theory of plasticity.’ Oxford University Press. 2.12 Wilshaw. T. R., Rau. C. A., Tetelman. A S., (1968). ‘A general model to predict the elastic-plastic stress distribution and fracture strength of notched bars in plane strain bending.” Engineering Fracture Mechanics, Vol. 1, pp 191~211. 2.13 Tetelman. A. S., McEvilly. Jr .A .J., (1967). ‘Fracture of structural materials.’ John Wiley and Sons. 2.14 Xu. R. X., Thompson. J.C., Topper. T. H., (1995). ‘Practical stress expressions for stress concentration regions’. Fatigue and Fracture of Engineering Materials and Structures, Vol. 18, N° 718, pp 885~895. 2.15 Lin K. Y., Pin. Tong., (1980). ‘Singular finite elements for the fracture analysis of V notched plates’. International Journal for Numerical Methods in Engineering, Vol. 15, pp 1343~1354. 2.16 Angot. G., Pluvinage. G., (1996).’Local strain fracture criterion for notched specimens’. Problems of Strength, Special Publication 96, pp.3~14.

CHAPTER 3 STRESS CONCENTRATION FACTOR ______________________________________________________________________ 3.1 Definition of the stress concentration factor 3.1.1 GROSS STRESS AND NET STRESS The presence at the notch tip of a maximum stress Vmax has been previously mentioned. It is traditional to evaluate this maximum stress relative to the stress which is applied to a non-notched specimen. Three cases can be considered tension, torsion and bending. Vyy

Vyy

Vmax

Vmax VN x

U

x

U

VN Vg

Vg

case 1 tension

case 2 and 3 torsion or bending

Figure 3.1: Definition of gross, net and maximum stress in a notched specimen in tension and torsion or bending.

Gross and net stresses are defined for the three modes of loading (tension bending and torsion) in table 3.1. where B is the thickness, M the bending moment, T the torque, I the inertia, J the polar inertia W width and b is the ligament size. This leads to two definitions of the stress concentration factor according to Peterson [3.1]. The first is relative to net stress: .

kt

V max . VN

(3.1)

The second is relative to gross stress: 41

G PLUVINAGE

42

kt

V max . Vg

(3.2)

The second formula for the stress concentration factor is the most commonly used. It should be noted that the term kt is devoted only for elastic stress. Loading mode Surface Gross stress Surface Net stress tension P P Vg VN BW Bb bending

torsion

Vg

M .W 2I

VN

M .b 2I

Vg

T .W 2I

Vg

T .b 2I

Table 3.1 Gross and net stresses on defined on surface for the three modes of loading

3.1.2 INFLUENCE OF NOTCH CONCENTRATION FACTOR

RADIUS

ON

ELASTIC

STRESS

It has been shown that the stress concentration factor increases when the notch radius decreases; sharp notches lead to higher maximum stresses than blunt notches. For example the stress concentration factor at the notch tip (versus the notch radius) in a CT specimen is presented in Figure 3.2.

Elastic stress concentration kt 50 CT SPECIMEN ; Notch angle < = 0°

40 30 20 10 0 0

0.2

0.4

0.6 0.8 1 Notch radius U (mm)

Figure 3.2: Influence of notch radius on the elastic stress concentration factor (CT specimen, notch angle 0°).

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

43

3.1.3 THREE EXAMPLES OF VALUES OF THE ELASTIC STRESS CONCENTRATION FACTOR Three examples of elastic stress concentration factor values are presented here to illustrate the influence of the geometry and loading mode : í a cylindrical notched bar in tension, í a notched plate in tension, í a notched plate in bending. Compendiums of many elastic stress intensity factors can be found in Peterson [3.1] and Roark’ formulas for stress and strain [3.2].

3.1.3.1 Cylindrical notched bar in tension The geometry of this cylindrical notched bar in tension is given in figure 3.3, the evolution of the elastic stress concentration factor with the notch radius in figure 3.4 and the associated analytical formula is given in (3.4). U

d0

D

Figure 3.3: Geometry of a cylindrical notched bar in tension.

2.5

Elastic stress concentration kt

2.2 1.9 1.6 1.3 1.0

1.0

1.5

2.5 3.0 2.0 Notch radius (mm)

Figure 3.4: Evolution of the elastic stress concentration factor with notch radius.

G PLUVINAGE

44

§b · 2¨¨  1¸¸. U © ¹ kt § §b · ¨¨  1¸¸. arctan g ¨ ¨ ©U ¹ © 3.1.3.2 Notched plate in tension or in bending

b

U b ·¸  U ¸¹

b

.

(3.4)

U

The geometry of a notched plate in tension or in bending and the evolution of the elastic stress concentration factor with the notch radius in figures 3.5 and 3.6; the associated analytical formula is given in formulas (3.5 and 3.6).

3.2 Elastoplastic stress and strain concentration factor Generally the elastic stress concentration factor is relatively high and the maximum stress exceeds the yield stress. In this case a plastic relaxation occurs and the maximum stress Vmax decreases. At the same time, the maximum strainHmax increases. Because the strain is not proportional to the elastoplastic stress, elastoplastic concentration factors are different. Plate in tension

Plate in bending

W W

W UU

a

aa

U

Elastic stress concentration factor kt 3.0

a/W 0.2 0.3 0.4 0.5

2.5 2.0 1.5 1.0 0.0

0.25 0.50 0.75 1.0 Non dimensional notch depth a/U

Figure 3.5 : Evolution of the elastic stress concentration factor with notch radius for a plate in tension .

Elastic stress concentration factor kt 2.0

a/W

0.2

1.75 1.5

0.3

1.25

0.4

1.0

0.6 0.8 0

0.25

0.5

0.75

1.0

Non dimensional notch depth a/U Figure 3.6: Evolution of the elastic stress concentration factor with notch radius for a plate in bending.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

plate in tension kt = K1 + K2* (a/W) + K 3 *(a/W) 2 K4 *(a/W) 3

(3.5)

K1 = 0,721 +2,394*¥(a/U -0,127*(a/U  K2=1,98 -11,489* ¥(a/U +2,211*(a/U K3= -4,413 +18,751*¥(a/U -4,596*(a/U  K4= 2,714 -9,655*¥(a/U +2,512*(a/U  plate in bending kt = K'1+K'2* (a/W) +K'3 *(a/W) 2+K'4 *(a/W) 3

(3.6)

K'1(x) =0,721 +2,394*¥(a/U -0,127*(a/U  K'2 =-0,426 -8,827*¥(a/U +1,518*(a/U  K'3 =2,161 +10,968*¥(a/U -2,455*(a/U K'4:=-1,456 -4,535*¥(a/U +1,064*(a/U  The elastoplastic stress concentration factor is defined by: kV

V max, pl

,

V g , el

45

 (3.5)

where the maximum stress is plastic and the gross stress is elastic. For the case where the gross stress is plastic, we can define a plastic stress concentration factor kp

kV

V max, pl

.

V g , pl

(3.8)

The elastoplastic strain concentration factor is equal to: kH

H max, pl H g , el

.

(3.9)

Numerous analytical solutions are available in the literature for the elastic stress concentration factor. Values of the elastoplastic stress and concentration factors depend on the material behaviour and load level. Hence they are generally obtained by Finite Element Method.

3.3 Relationship between the elastic and elasto-plastic concentration stress and strain concentration factors and the elastic one In order to generate solutions for stress concentration factor, several authors have proposed relationships between the product of the elastoplastic stress and strain concentration factors and the square of the elastic stress concentration factor.

G PLUVINAGE

46

V max H max . Vg Hg

kV . k H



f k t2 .

(3.10)

In the following paragraph the parameter ] is used to evaluate this relationship: k t2 [ . (3.11) kV .k H 3.3.1

NEUBER’S METHOD [3.3]

The basic assumption of Neuber’s rule is that the area B is the product of the area A, as given in figure 3.7, by the square of the elastic stress concentration factor A V g . H g ,  B k t2 . A      B V max . H max k t2 .V g . H g  with    Stress Vmax,el B = Vmax.Hmax B = (kt)2.Vg.Hg A = Vg.Hg Vmax

B

Vg A

Strain Hg

Hmax

Figure 3.7: Principle of Neuber’s method. Definition of the elastic stress concentration within areas A and B.

Elastoplastic stress and strain concentration factors are defined by: 

kV

By writing Neuber’s assumption:

V max , kH Vg

H max . Hg

 

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS



V max . H max

k t2 V g . H g 

 



V max .H max V g .H g

k t2 

 



V max H max . Vg Hg

k t2 

 

47

This leads to Neuber‘s rule: kV . k H

and

9

* Validity of Neuber‘s rule

k t2 2

(3.18)



 

 kt

kV .k H

íElastic case By definition, the maximum and gross strains are equal to: 

H max

V max

 H g

E The elastoplastic strain concentration factor is  V max E kH ˜  kV E Vg

Vg E



k t 

 

 

Again we find Neuber‘s rule kV . k H The basic assumption is verified:



k t2 .

V max .H max k t2 .V N .H N 

(3.22)

 

 íElastoplastic case The material is assumed to follow the Ludwik’s law V = KHn where n is the strain hardening exponent and K the hardening coefficient. From Ludwik’s law, we get the maximum strain 1 V max n , (3.24) H max K kV V g n , (3.25) and H max K By multiplying by Hmax : V 2g  1 k V V g .H (3.26) H nmax max K .E . k V k H K



The product kV.kH is equal to:

G PLUVINAGE

48

kV .k H

1 K .E.H nmax V 2g

(3.27)

and n  1 K .E. H n .H K .E.H max V max , el max max V max .V max , el , (3.28) kt. V 2g V 2g V 2g Vg Vmax,el is the elastic maximum stress defined in figure (3.2) and Vmax,el > Vmax V max , el (3.29) kt. k V . k H ! k t2 . Vg We can notice that Neuber ‘s rule for the elastoplastic case underestimates the product kV.kH * modification of Neuber’s rule Yi Sheng Wu [3.4] has proposed a modification of Neuber’s rule based on a material parameter m. Neuber’s relationship is thus modified as follows: (3.30) C1m . C12 m 1 , with kV kH ,C2 . (3.31) C1 kt kt The parameter is expressed by: m = 0.49+0.31n - 9.17(K/E), in plane stress conditions and in plane strain conditions m m' 1 Q 2





(3.32) (3.33)

where n and K are respectively the hardening exponent and coefficient of the Ludwik’s law of the material and E Young’s modulus. This relationship is differs little from plane stress to plane strain conditions. When m = 0.5, we obtain again Neuber’s rule. For the elastic case:

for elastoplastic case :

kV

kH

kt ; ; C1 C2

1,

kV  kt ; k H ! kt ; C1  1 ; C 2 ! 1 .

3.3.2

(3.34) (3.35)

MOLSKIíGLINCKA’S METHOD [3.5]

MolskiíGlincka’s method considers the strain energy density at the notch tip W*.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

H max ³ V H max dH .

W*

49

(3.36)

0 The net strain energy density is equal to :

Hg ³V g H g d H g .



W *0

(3.37) 0 The elastic stress concentration factor is related to the square root of the ratio of these two strain energy densities: W* . (3.38) kt W *0 It is assumed that the material undergoes strain hardening and the stress- strain curve consists of elastic and plastic regions: n

V = EH + KH .

(3.39) E is the Young’s modulus, K the strain hardening coefficient and n the strain hardening exponent. Stress Vmax = kV .Vg

H

g

³V §¨©H

A

g

0

H B

Vg

g

Hg

Hmax = kH.Hg

t

H

g

max

³V §¨©H 0

k

·¸.d

g¹

·¸.d

g¹

H

g

B A

Strain

Figure 3.8: Principle of Molski-Glincka’s method. Definition of the elastic stress concentration within areas A and B.

The strain energy density at the notch tip W* is: H max H max 1 E H 2max K H nmax W*  , ³ VdH ³ EH K H n dH n 1 2 0 0



or



W*

n 1

1



V 2max V max V max  . 2

(3.40)

K

n

.

(3.41)

Assuming that the gross stress remains elastic, the strain energy density is equal to :

G PLUVINAGE

50

V 2g

W 02

2E

.

(3.42)

Relationship (3.42) leads to: k t2 .

V 2g 2

2

n 1

1



V 2max V max V max  . K

n

. (3.43)

By extension to the case for Re < Vg 1 º ª n» « V 2g V g V g 2 « » .  . kt K n 1 « 2 » »¼ «¬ x Expression for ] We define the ]parameter as:

§¨ ©

2

k t2 kV . k H

n 1

1



V 2max V max V max  .

·¸ ¹

K

n

.

[

(3.44)

(3.45)

The value of ] can be obtained for the following case: Re < Vmax Re < VgRe > Vg; x For Re < Vmax : ] =1 ; For Re < Vg : we introduce the products Vmax.Hmax and Vg.Hg

x

k t2 . k t2 .

k t2 .

V g .H g 2 V g .H g 2

V g .H g 2

V max .H max V max 2



n 1

V max .H max .§¨ 1  ©2

. H max ,

1 · ¸, n 1¹

n2 · ¸¸ , 2 © n  1 ¹

V max .H max .§¨¨

(3.46) (3.47)

(3.48)

k t2

· §V ¨ max . H max ¸.§¨ n  2 ·¸ , ¨ V g . H g ¸ ¨© n  1 ¸¹ ¹ ©

(3.49)

k t2

§ n2 · ¸¸ . k V . k H .¨¨ © n  1 ¹

(3.50)

This leads to:

[

§ n2 · ¸¸ . ¨¨ © n  1 ¹

(3.51)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

51

if Re < Vg : ª V .H º g g Vg  .H g » k t2 .« « 2 » n 1 ¬ ¼

ªV º .H V « max max  max . H max » , 2 n 1 «¬ »¼

ª 1 ·º §1 ¸» k t2 .« V g . H g .¨  2 n  1 ¹¼ © ¬

ª 1 ·º §1 ¸» , «V max . H max .¨   1 ¹¼ 2 n © ¬ V max .H max . k t2 V g .H g





(3.52)

(3.53) (3.54) (3.55)

] = 1 and we found again the Neuber’s rule. 3.3.3

STOWELLíHARDRATH-OHMAN’S METHOD [3.6]

In the StowellíHardrathíOhman’s method, the elastoplastic stress concentration factor is given : kV 1 E s , (3.56) E k t 1 E 1  k t  1 . s , E where Es is the secant modulus defined by:

or

kV

V max . H max

Es

The elastoplastic strain concentration factor is given: Es , kH kH . E

(3.57)

2

k H ˜ kV

Es º Es ª «1  k t  1 ˜ E » ˜ E , ¬ ¼ 3

Ÿ k H ˜ kV

3.3.4

2

Es §E · §E ·  k t  1 2 ¨ s ¸  2k t  1 ¨ s ¸ . E © E ¹ © E ¹

(3.58)

KOE’S METHOD [3.7]

In Koe’s method the basic relationship between the elastic, the elastoplastic stress concentration factor and the elastoplastic strain concentration factor is given by :

G PLUVINAGE

52

§ k t 1 · § k t 1 · ¨ ¸ ¨ ¸ ¨ k  1 ¸  1 2,4¨ k  1 ¸ . © V ¹ © H ¹

(3.59)

This leads to:

] 3.3.5

2,4 ˜ k t ª k H  k t ˜ kV k t º k t ˜« . » kt  1 ¬ k H ˜ kV ¼ kH

(3.60)

MAKHUTOV’S METHOD [3.8]

Makhutov has postulates that:

k t2 kV ˜ k H

f k t , V N , Re, n * .

(3.61)

The function f is expressed versus the ratio of the maximum stress and the yield stress. n* k t .V N , Re where the exponent n* is given by : 1  n ª § V N 1 ·º n* .«1  ¨  ¸» . 2 ¬ ¨© Re k t ¸¹¼ f

3.3.6





(3.62)

(3.63)

MOROZOVíPLUVINAGE’S METHOD [3.9]

Let us consider the following non-linear elastic problem concerning the concentration near the notch. The notch is located in a planar structure loaded on the boundary by known forces. The part contains a triangular notch with circular root. The radius of curvature is U, the angle between opposite sides is \. The sides are assumed as straight lines. The stressístrain diagram of the constitutive material presents a power deformation law :  V = .Hn. (3.64) where . and n are the strain hardening coefficient and exponent. Due to the monotonic increase of external load and the absence of unloading, the non linear elasticity and deformation theory coincide. This is the consequence of an elastic potential defined as the stain energy density. W*

H

³ V ijd H ij 0

(3.65)

3.3.6.1 M Integral To solve the problem reduced as an elasto-plastic stress and strain concentration problem, let us introduce the following integral:

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

M

ª § wu j ·º  2N u j ¸» n k ds , ³* «W * x k V ij , k .¨¨ x l ¸» «¬ © w xl ¹¼

53

(3.66)

where i, j, k, l = x,y ; W* is the strain energy density, xk are the co-ordinates, the product Vik.nk corresponds to the traction on the path *and uj is the displacement.

Load

N

Displacement Figure 3.9: Definition of the N ratio

We define a coefficient N which can be seen on the stress strain diagram (figure 3.9) by the ratio of two areas. The product NVij.Hij is the dashed area which is the difference of the areas under the deformation diagram and the straight line of the linear elastic loading. From this picture, we have found the useful expression. 2W* = (1 + 2N). Vij . Hij

(3.67)

It can be demonstrate that this integral is equal to zero along a closed path *.Using Gauss's theorem: wF d: (3.68) ³ n i.F x k ds ³³ w * : xi :is the surface included by the path *

G PLUVINAGE

54

y *1 A D

*2 B C

x

0

Figure 3.10: Definition of the path integral.

From line integral transformation, we have the following expression for M integral. § wu j ·º w ª (3.69) «W * x k V ij , k .¨ x l M ³*  2N u j ¸»dxdy . ¨ wx ¸» w xk « l © ¹¼ ¬ Here the first integrand term is: w W * . x k wW * . x  2W * . (3.70) w xk w xk k The second integrand term is zero from equilibrium conditions w ª

wu j

º

 2N u j » «x V jk w x k ¬« l w x l ¼» wu j w xl w u l wN w V jk , w w u jl  2N V jk x l  2N u j  V jk x l V jk w x k w xl w x k w xl w xk w V jk w x k

1  2N V jk xl

wu j w xk

xi

wu j w xi

 V jk x l w

w2u j x k w xl

,

1  2N V jk H jk  x k wW * .

(3.71)

w xk

Here we use the following expression wu j

V jk w xk

wu j 1 wu j 1  V jk V jk 2 w xk 2 w xk

§ wu j wu · 1 k¸  V jk ¨¨ 2 w xk w x j ¸ ¹ ©

V jk H jk .

(3.72)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

55

Substituting both terms into (3.69) and taking into account (3.67), we find that this line M integral is equal to zero for a closed path. It can be shown that this integral is pathindependent. Choosing the integration path in the form as shown in figure 3.10, the part of the integral coincides with the sides of the notch AB and CD. It is equal to zero because xknk = V jk.nk = 0 and these conditions are a consequence of the presence of a free boundary and the localisation of the origin of co-ordinates at the corner of the notch. For a closed path M*1 + M*2 = 0, so the M integral is path independent. Due to the absence of analytical solutions for strains and displacements, we will introduce a number of assumptions to obtain the complete results. 3.3.6.2 M-integral value for notch problem Let’s coincide the integration curve *1 with the body boundary in the range (-D, D) of a circle arc at the notch root (figure 3.10) and introduce the notation: S \ 1  S \ . (3.73) D 2 2 2 Given that the boundary of the notch is free from load, the M1-integral is:

³ W * xdy  ydx , *1 § dy · § dx · x¨ ¸  y ¨ ¸ . © ds ¹ © ds ¹ M1

xk nk

xnx y ny

(3.74) (3.75)

If we also assume that:

W * W *n . cos T . For a crack \ = 0, the J integral is reduced to: 1 J SU V 2max , 4E where E is the Young’s modulus.

(3.76) (3.77)

The strain energy density W*m near notch root for T= 0 may be easily calculated in the case of uniaxial tension: W *m

³ VdH

V * ³ H ndH

V * 1 n .H , 1  n max

a ST 2 dT , M 1 W *n U . ³ cos 2T  E cos T  sin 2T cos 2D a



Therefore



(3.78) (3.79)

G PLUVINAGE

56

M1

§ · 2D 2  n ¨  2DE sin \  4D . 1max  ...¸ cos 2 D  ¨ \ ¸ 2 S 1 n S © ¹



V*

2 · § ¨ ...  4 D sin S  4D  4D sin 2 D ¸ U 2 , ¸ ¨ 2D S  4D S  4D © ¹

(3.80)

Choosing a second integration contour *2 coinciding with the rectangular boundary of the specimen as shown on figure 3.10, we can present the M-integral in a more recognisable form: M

M

§ wu j ·º w ª  2N u j ¸»dxdy , «W * x k V ij , k .¨ x l ¨ ¸» x k «¬ © w xl ¹¼ § wu j ·  2N u j ¸dx , ³* W * xdy  ydx  ³ V jxdy  V jy .¨¨ x l ¸ w xl © ¹ * ³* w

(3.81)

ª º ª º § wu j · § wu j ·  2N u j ¸ V jy »dx  «W * x  ¨ x l  2N u j ¸ V jx » dy . ³ « W * y  ¨¨ x l ¸ ¨ wx ¸ »¼ »¼ l © w xl ¹ © ¹ *¬« ¬« E

D

y

U <

F

*2

x

A a

E.b W

D

C

Figure 3.11: Notched specimen and contour *2.

(3.82)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

57

Considering the M-integral on each side of the specimen separately, on sides AB and EG we have: (nx = -1 ; ny = 0 ; x = -a ; dx = 0 ; Vjx = 0 ; W* (x,y) = W*(-a,y)). It can therefore be proposed that:

M AB, EG

M AB, EG

yb yb  W * dy a ³ ³ W * dy ya ya §L · 0 ¸ ¨ 2 aW * ¨ ³ dy  ³ dy ¸ ¨ 0 L ¸ 2 ¹ © W*

(3.83)

W * La ,

(3.84)

V2 ,

(3.85)

2E

W *AB W *EG W *BC W *CD W *DE W * ,

(3.86)

because the sides are not disturbed by the notch. Suppose that: wu y ux=0,uy =Hy and 0. wx We have on sides BC and DE nx= 0 ; ny =± L y

M BC

M DE

rL

2

; V yy V xy

wu y 1 1 1 , W * Lb  Lb1  2N .V .H  Vb.b  2a 2 2 2 wx xB § wu y L L· L  H  2NH ¸dx ³  W * dx  V ¨¨ x 2 2 2 ¸¹ w x © xD

.

W * Lb LbVH 1  2N w u y   bb  2a . 2 2 wx On side CD (nx = 1 ; ny = 0 ; x = b - a ; dx = 0, Vjx = 0) L/2 M CD ³ W * b  a dy W * Lb  a . L/2 Taking into account expression (3.91) the M-integral is given as: M DE

(3.87) 0

(3.88)

(3.89)

(3.90)

(3.91)

G PLUVINAGE

58

M

M2

M AB  M EG  M BC  M CD  M DE

V

wu y wx

bb  2a ) .

(3.92)

y

e x

o x0 a

E.b

W-a-E.b

Figure 3.12 : Strain distribution at notch tip.

3.3.6.3 Values of the ] factor Deformation in the net section ligament leads to a rotation of the upper part of the specimen with respect to the lower part. The rotation angle T resulting from this deformation is equal to the partial differentiation of displacement uy with respect to x. Given this, we have to consider the nominal stress and strain for three cases: i) VN and Vmax < Re. ii VN < Re.< Vmax iii) VN > Re.> Vmax í case i) VN and Vmax < Re We have the Neuber expression: kV k H 1, (3.93) k t2 where kt is the theoretical stress concentration factor for linear elastic material. The stress and strain distributions may be found with the use of the plain section hypothesis and equilibrium equations.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

VN

P  B.W  a )

6Mb B. W  a )

2

.

59

(3.94)

P = VgBW and the bending moment Mb= Pe (e is the eccentricity of load).

VN

. (3.95) E a is the depth of the notch, B the thickness of the specimen and E the Young’s modulus. 1 e W  a  EU . (3.96) 2 From the plane section hypothesis, the net-section strain in the y-direction is equal to H = Ox + c while the stress is equal to V = KHn (. and n are the material constants). For x = x0 , the strain is zero H0 = 0 and x0 = -c/O. (figure 3.12) The equilibrium conditions for the forces and the moments are : W 1 Z ; : OZ  c ; :* OZ  c , (3.97) 2

HN

P

Z

³ VBdx

Z

Z n kB ³  Ox  c dx Z

Z ³ VBdx Z

Mb

Z

1  nº kB ª , « Ox  c » O 1  n ¬ ¼ Z

Z n k .B ³  Ox  c dx , Z

(3.98) (3.99)

Z

2n 1  nº c kB ª Ox  c Ox  c   2n 1 n » , O2 « ¼ Z ¬

 OZ  c

1 n

   OZ  c

:1  n  : *1  n

1 n

O 1  n B.k

(3.100)

P,

O 1  n

P, B.k 2n 2  nº 1 ª    OZ  c « OZ  c » 2n ¬ ¼

1 n 1  nº c ª    OZ  c « OZ  c » 1 n ¬ ¼

O2M

,

B.k

1 c O2M , (3.101) : 2  n  : *2  n  :1  n  : *1  n 2n 1 n B.k From this system we can findO and c. The net strain and stress are then given by:

>

@

>

@

G PLUVINAGE

60

W a c, 2 n W a k. O c . 2

OZ  c O

HN



VN



(3.102) (3.103)

As previously defined, the rotation angle T owed to deformation in the net section layer is equal to T = ˜Uy/˜x. From the geometrical condition we find that: ³ H N dy 0 Z  x0

T

HN

VN E

; if VN < Re ; H N

[UH N Z  x0

(3.104)

OZ  c if VN >Re.

where ] is an empirical coefficient and: w u y [U H N wx

Y  x0

.

(3.105)

(3.106)

Given that M1 = M2we find: [U H N k 1 § · . H 1  n . U 2 ¨ 2 sin D  ED  E sin 2D ¸ V .W .W  a . 2 1  n max Z  x0 © ¹

(3.107)

We can now consider the deformation Hmax near the notch tip for two last cases, taking into account the definition of the elastoplastic concentration factors kVand kH for elastoplastic region near a notch. kV

V max ; kH VN

kH

H max HN

H max . HN

(3.108)

* Case ii VN < Re < Vmax

H max .E . VN

(3.109) (3.110)

From this it follows

H max k H H max

n 1

VN E

,

k H .kV .

(3.111)

V 2N KE

.

(3.112)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

61

* Case iii)VN and Vmax > Re Suppose now that the net section stress is greater than the yield stress, then in this section : V =K . Hn. 1 H max H max K n , (3.113) kH 1 H Nx V Nn

kV

 H max

and

n Therefore K H max

K . H max

V max VN n

1

K N

.

K

,

VN kV

1

VN N VN

n

VN K

VN

.kV .k H

K

.

(3.115)

1 n

1 n

(3.114)

N

N

. kV .k H .

(3.116)

Now we may write VN > Re and Vmax > Re

UV N 1 n

. f D , E . k V . k H

[ .W  a P . . Z  x0 B

(3.117)

In finding the coefficient [ we use Neuber's relationship and the maximum strain energy density: EV 2 max . (3.118) W *m 2 .

In this case, n = 1, K = E, kH = kV = kt and for uniformly applied stress, the distance x0 is : 2  W  l . (3.119) x0 6l and

M1

M1

EV 2 max . U 2 . f D , 2

(3.120)

V NHN

(3.121)

2

. k t2 . U 2 . f D .

From (M1 = M2), we can deduce the value of [.

G PLUVINAGE

62





ª Z  x0 º . k t2 . f D .B.« », 2 «¬ P.W  a »¼ 2 º ª «  W  2l » U 2 [ . k t . f D .B.« ». 12 . W  2a » « l.W  l  ¬ ¼ So, finally these expressions may be written as: k t2 í for Vnand Vmax < Re [; kV . k H

[

VNU

(3.122)

(3.123)

(3.124)

í for Vn < Re and Vmax > Re k t2 kV . k H

1 n ; 2

k t2 kV . k H

1 n K . 2 VN

(3.125)

í for Vn and Vmax > Re



1

n 1 n .H N .

(3.126) This third solution is the only analytical solution for VN and Vmax > Re. 3.4

Evolution of elastic and elastoplastic stress (or strain) concentration factor with net stress

The evolution of kV and kH versus the net stress section is given y: í for Vn < Re and Vmax > Re 1 1  n  1  n 1 kH , . 2 k t k t 1  n 1  n n 1  n  1  n 1 kV , . 2 k t k t 1  n 1  n

(3.127)

(3.128)

í for Vn and Vmax > Re kV kt

n 1  n

kt

 1  n 1 . 1  n 1  n 2

.



K 1 n n .H N , VN

(3.129)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

kH kt

kt

 1  n 1 . 1  n 1  n 2

1 1  n .

  K

1  n 1  n / n

VN

.

63

(3.130)

This kind of evolution is presented in figure 3.13 for a CT specimen with a notch radius equal to 1mm. For VN/Re<0.2, kV decreases with and kH increases. Elastoplastic stress and strain concentration factor kV ,kH 20 15

CT SPECIMEN Notch radius U = 1mm ke kV

10 5 0 0.0

0.2

0.4

0.6

0.8

1.0

Non dimensional net stress VN/Re Figure 3.13 / Evolution of the elastic, elastoplastic strain and stress concentration factors with ratio VN/Re.

3.5 Comparison of ] evolution with net stress

The relationship between stress and strain concentration factors is sensitive to the constitutive equation V = f(H) and the definition of the coefficient. k t2 ( 3.131) [. kV . k H For case where V1 < Re and Vmax > Re in which [ varies according to the expression derived by various authors (see table3.2). Relationships which express] = kt2/(kHkV) as a function of the net stress = f(VN) have been compared with results obtained from Finite Elements results using a CT specimen made with a material having a strain hardening exponent n = 0. 065. The results of this comparison are shown in figure 3.14. It is important to note that these formulae are only valid if the net section stress is less than the yield stress with the exception of the MorozovíPluvinage formula. For this reason the comparison is limited to VN ” Re. From figure 314, it can be seen that Makhutov and Stowel‘s solutions are close to FE computed results. Those of Glincka and MorozovíPluvinage solution are higher and very close together because the strain

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64

hardening exponent is low. It is also noticeable that when the net section stress exceeds the yield stress, there is a wide variation away from the computed results. Authors

Expression  ] =1

Neuber [3.3]

Molski-Glincka [3.5]

For Re < Vmax : ] =1 For Re• Vmax ] =

n n 3

Stowell, Hradrath and Ohman [3.6]

[

Es §E · §E ·  k t  1 2 ¨ s ¸  2k t  1 ¨ s ¸ E © E ¹ © E ¹

Koe, Nakumara and Tsunemari [3.7]

]

2,4 ˜ k t kt  1

2

ª k  k t ˜ kV k t º k t ˜« H » k H ˜ kV ¬ ¼ kH

n* k t .V N Re 1  n ª § V N 1 ·º .«1  ¨  ¸» with n* 2 ¬ ¨© Re k t ¸¹¼

[ Makhutov [3.8]

Morozov-Pluvinage [3.9]



For Re < Vmax : ] =1 For Re• Vmax [

2 n  1

Table 3.2 Different formulae for the expression ]

The influence of the notch radius on the value of ] has been studied and compared to the results obtained from finite element studies. Using a CT specimen with a notch angle \ = 40° and a net stress VN = 317 MPa (see figure 3.15), it can be seen that the computed results decrease with notch radius U and tends to unity when U increases. [ lies between the two limits: [ = 1 (Neuber) and [ = 1.65 (MorozovíPluvinage ) Finite element computation gives values of [ close to that given by Koe.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS ] parameter

] parameter

2.5

MorozovPluvinage

1.6

2.0 1.5

Makhutov 1.2

FEM Koe Neuber

1.0 Glincka Stowell Koe Neuber

0.5 0.0

65

0

100

200

FEM Makhutov Morozov

300

0.8 0.0

500 400 Net stress (MPa)

Figure 3.14: Evolution of ]versus net stress VN. Comparison of different formulas with computed FE results.

0.5

1.0

1.5

2.0 Notch radius U (mm)

Figure 3.15 : Evolution of [ parameter versus notch radius; (CT specimen, \ = 40°, VN = 317 MPa).

3.6 Conclusion If the maximum stress at the notch tip remains elastic its value can be obtained from the elastic stress concentration factor, a term that is very sensitive to notch radius. If this maximum stress becomes plastic, stress relaxation occurs and the elastoplastic stress concentration is less than the elastic one. However, the strain concentration factor is greater than kt. These three concentration factors are related in a relationship which depends of the level of the net section stress.

REFERENCES [3.1]

Peterson .R. E, (1974).’ Stress concentration Factors’; John Wiley and Son, New York, [3.2] Young. W .C . (1989). ‘ Roark’s formulae for stress and strain’, MacGraw Hill Internatinal Edition. [3.3] Neuber. N , Weis .V.(1962). Trans ASME, Paper N° 62-WA-270, [3.4] Yi Sheng Wu.(1988). ‘The improved Neuber rule and low cycle life estimation’,ASTM STP 942, pp 1007~1021. [3.5] Molski .K, Glinka. G. (1981). ,‘A method of elastic-plastic stress and strain calculation at a notch root’, Materials Science and Engineering, N°50, pp 93~100. [3.6] Stowell .H. Hardrath. H .F and Ohman. L. (1951). ‘ A study of elastic and plastic stress concentration factor due to notches and fillets in flat plates’ , NACA TN 2566 , December. [3.7] Koe. S. Nakumara. H. Tsunemari. T. (1978). Journal of the Society of Materials Science, Japan, Vol. 27, N°300, pp 847~852. [3.8].Makhutov.N.A, (1981).’ The deformation fracture criteria and strength calculation of the machine parts’. Mashinostroyenie, Moscou, , p 272. [3.9] Morozov. E. Pluvinage.G. (1996).‘Study of strain concentration coefficients by path-independant integral’.International Scientific & Technical Journal, Problems of Strength.

CHAPTER 4 CONCEPT OF NOTCH STRESS INTENSITY FACTOR AND STRESS CRITERIA FOR FRACTURE EMANATING FROM NOTCHES

4.1 Introduction An ideal and mathematical crack is considered as a simple cut in a plane. This definition leads to a zero notch radius or infinite acuity. The stress distribution near the tip of such a cut is characterised by a stress singularity, stresses have in theory an infinite value at the crack tip. For a non zero notch radius, the maximum stress has a finite value. However, it has been noted that the fracture process is governed only by point stresses at the crack or notch tip. This phenomenon requires a physical volume called the fracture process volume. This assumption is supported by fracture being sensitive to geometry, scale effects, and loading mode, even for the same maximum stress value. Limits of this fracture process volume are difficult to determine and change with the notch radius. For a simple crack it can be related to a microstructural unit (e.g. grain size, bainite lath size, etc...). For a notch this volume depends on the notch radius, geometry, and loading mode. Consequently, the effective fracture stress is also not so easy to define. A way to overcome these difficulties is to consider the stress distribution at a distance ahead of the crack or notch tip up to the limit of the fracture process volume. Beyond this limit, a constant relationship between stress and distance can be found. This classical linear fracture mechanics relationship is given in equation (4.1): 

V ij r

cst 

(4.1)

where Vij is the stress tensor, r polar co-ordinate. In addition, the probabilistic fracture criterion is given in equation (4.2): (4.2) V mww .V q cst , where Vw is the so called Weibull stress, mw the Weibull modulus and V0 the elementary volume. Such fracture criteria are called associated parameters fracture criteria. In the context of the linear fracture mechanics analyses, the constant for equation (4.1) is equal to K/— 2S where K is the stress intensity factor. This probabilistic fracture criterion is given in equation (4.2). Similarly as in fracture mechanics, the Notch Stress Intensity Factor can be used as a fracture criterion of associated parameters. However if a procedure to determine the size of the fracture process zone is available, a two parameter local fracture criterion can be used 66

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

67

for fracture emanating from notches. These two parameters are respectively the size of the diameter of the fracture process volume (assumed cylindrical) and the effective fracture stress. The application of this fracture criterion allows the mechanical property know as fracture resistance or fracture toughness. This mechanical property is sensitive to temperature and loading rate because the fracture process requires local plasticity to promote fracture initiation and plasticity is a thermal activated process. Fracture toughness is also sensitive to hydrostatic pressure and consequently to notch radius which affects the value of triaxiality. A fracture toughness transition associated with notch radius can therefore be found. 4.2 Concept of stress intensity factor 4.2.1 STRESS INTENSITY FACTOR FOR A CRACK [4.1] Linear fracture mechanics assumes that the stress distribution near the crack tip for opening mode of fracture (mode 1) is governed by the following relationship:

V ij

KI f ij T , 2Sr

(4.3)

where KI is the stress intensity factor and fij (T) is an angular function.

Vyy

y

Vxx Vyx

r crack

T

x

Figure 4.1 : Representation of a bi-dimensional plane stress condition near a crack tip.

In order to precisely determine this angular function, for a bi-dimensional plane stress condition, the three components of the stress tensor are required see figure 4.1and equation 4.4. :

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68

V xx T

3T T § T KI cos .¨1  sin sin 2 2 2 2Sr ©

V yyT

3T · T § T KI cos .¨1  sin sin ¸ , 2© 2 2 ¹ 2Sr

V xyT

3T · T § T KI cos .¨ sin sin ¸ . 2 ¹ 2© 2 2Sr

· ¸, ¹

(4.4)

The relationship between the distribution of the normal stress and the distance (in a nondimensional form) is shown in figure 4.2. Based on this distribution the definition of the (crack) stress intensity factor is given in equation 4.5: lim § · o 0 ¸ V yy r , T . r . (4.5) 2S .¨ r  KI © ¹ Log (Vyy/Vg) 10 CRACK

Stress singularity 0.5 1

1

1 10-4

10-2

1

Log (r/B)

Figure 4.2: Distribution of the stress normal to the crack plane divided by the gross stress presented in a bilogarithmic graph where B is the thickness.

The distribution of the stress normal to the crack plane (which plays an essential role in the opening mode of fracture) divided by the gross stress can be presented in a bilogarithmic graph as a simple line of slope 0.5. The stress intensity factor KI gives the position of this line.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

69

4.2.2 STRESS INTENSITY FACTOR FOR A NOTCH WITH AN INFINITE ACUITY

4.2.2.1 Williams’s solution [4.2] In the case of a notch of infinite acuity (i.e. notch radius U = 0; notch angle < z0), Williams [4.2] proposed a solution for the stress distribution at the notch tip. Here, we define J angle as J S< (4.6) The radial, circumferential and shearing stresses are given by: K *I .^ cos>D  2 T @. cosDJ  cos>D  2 J @cos DT ` , V TT C I 2Sr D D 2 ½ ­ K *I .®cos>D  2 T @. cosDJ  cos>D  2 J @cos DT ¾ , V rr  D D 2  ¿ ¯ C I 2Sr

V rT

K *I

D

­

½

.®sin>D  2 T @. cosDJ  cos>D  2 J @cos DT ¾ .(4.7) D 2 ¿ C I 2Sr D ¯

y notch

Vrr

VrT

with r

infinite

VTT T

x

< J Figure 4.3: Representation of a bi-dimensional plane stress condition near a notch of infinite acuity in polar co-ordinates.

C1 is the first root of the equation: cosD  2 J  cos DJ

0

(4.8)

D is a coefficient less than 0.5 (D ” 0.5) and varies with notch angle according to :

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\ 2 \ 3 §\ ·  0.853 ¸  0.442 S S ©S ¹



D \ 0.5  0.089¨

10



(4.9)

Log Vyy/Vg NOTCH WITH INFINITE ACUITY

D< 0.5 Stress singularity

1

1 10-4

10-2

1

Log (r/B)

Figure 4.4: Distribution of the stress normal to notch plane divided by the gross stress presented in a bilogarithmic graph, B is the thickness.

The notch stress intensity factor for a notch with an infinite acuity K*I is given by: D§ lim · (4.10) o 0 ¸ V TT r ,\ , T 0 . r D K I  2S .¨ r  © ¹ The distribution of the stress normal to the crack divided by the gross stress can be presented in a bi-logarithmic graph as a simple line of slope D. The stress intensity factor K*I is also represented by this line (figure 4.4).

4.2.2.2 Gross and Mendelson’s solution [4.3] Gross and Mendelson have define the mode I and mode II notch stress intensity factor in a similar way : lim · § 2S .¨ r  (4.11) o 0 ¸ V TT r , T 0 . r D , K *I ¹ © lim · § 2S .¨ r  o 0 ¸ V rT r , T 0 . r D . (4.12) K *II ¹ © However, the value of the stress intensity factor is different because the constant (2S)D is replace by —2SThe two definitions therefore coincide only for <= 0.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

71

Singularity coefficient D 2

D < = 0,5 - 0,089

0.5

< S

 

< S

3

 

< S

0.4 0.3 0.2 0.1 0 0

45

90

135 180 Notch angle (°)

Figure 4.5: Evolution of the exponent D with notch angle< according to Williams [4.2].

4.2.2.3 Hasebe’s solution [4.4] For a short V crack, Hasebe expresses the circumferential stress distribution from the max

maximum stress

V TT \ ,T

V TT

where the stress is given at the distance U/2:

max \ , T 0 CT .V TT

0 . r D .

(4.13)

Here CT and D are parameters which depend on the notch angle \ according to figure 4.6. The notch stress intensity factor is given by :

K *I U

0;\

0

lim · max .¨ U  \ ,T o 0 ¸ V TT 2 © ¹

S §

0 . r D .

(4.14)

4.2.2.4 Lin and Pin Tong’s solution [4.5] Lin and Pin Tong have determined using a finite element method the stress distribution in a simple edge notch plate loaded in tension. 5 or 9 node iso-parametric special

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elements of size O have been used to determine the stresses via the computed displacements.

1.5

Parameter D

Coefficient CT

0.6 CT D

1.3 1.1

0.4

0.9 0.7

0.2

0.5 0.3

0.0

0

45

90

135

180 Notch angle (°)

Figure 4.6: Evolution of CT and D parameters with notch angle \ according to Hasebe[4.4].

y 3

2

1

4

T 2<

5

x

O Figure 4.7 : 5 nodes isoparametric special elements used by Lin and Pin Tong [4.5].

The notch stress intensity factor K*, is given by:

73

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

K *I

V yy 2S .D >1  D  D cos 2T  cos 2DT @. O D .

(4.15)

In this case, the stress singularity is of 1/rD dependence and varies in the range [0.5~0.45] when < is varied from 0° to 45 °. 4.2.2.5 Knésl’s solution [4.6] Parameter q

1.3

Parameter D q D

1.1 0.9 0.7 0.5 0.3

0

40

80

120

160

Notch angle < (degree) Figure 4.8: Evolution of coefficients q andD with notch angle < according to Knésl’s solution.

From the Williams’s solution, Knésl has expressed the stress field at the notch tip as :

V rr

K I U

0,\ 2S

K I U

0,\

0

>





@



.h\ . r  D . 2  D  D 2 cos DT  q 2  3D  D 2 cosD  2 T ,

0

> >





@

@

.h\ . r  D . 2  3D  D 2 cos DT  q 2  3D  D 2 cosD  2 T , 2S K I U 0,\ 0 .h\ . r  D . D cos DT  q 2  3D  D 2 sin D  2 T . . (4.16) V rT 2S The notch angle function h (<) is given by the following relationship:  0.5 2 XD c h\ (4.17) 2  D 1  q

V TT



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where q andD are coefficients which are dependent of the notch angle < according to Figure 4.8and Xc characteristic distance. 4.2.2.6 Summary Table 4.1 presents a summary of the different formulae derived to determine the Notch Stress Intensity Factor for a notch with infinite acuity. All solutions lead to a stress singularity with an rD dependence. The solution given by Lin and Ping Tong is different and has been establish for a small notch angle where D is near 0.5.

Authors

Singularity

Formulas

IRWIN 1972

V ij

WILLIAMS [4.2] 1952

V ij

KI f ij T crack 2Sr K* C I  2Sr

D

f ij T ,\ , D

r

- 1/ 2

r

-D

notch with infinite acuity

GROSS and MENDELSON [4.3] 1972

HASEBE [4.4] 1978

LIN and PIN TONG [4.5] 1980

KNESL [4.6] 1991

K* f ij T ,\ , D 2S r D notch with infinite acuity

r

K* f ij T ,\ , D 2 rD notch with infinite acuity

r

-D

r

- (1- D

r

-D

V ij

V ij

V ij

K* f ij T , D .hO 2S r1  D

-D

. notch with infinite acuity K* f ij T , D .h\ 2S r D notch with infinite acuity.

V ij

Table 4.1: Summary of the different formulae for stress singularity.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

75

4.3 Concept of notch stress intensity factor

4.3.1 CREAGER AND PARIS’S SOLUTION [4.7] According to Creager and Paris, the elastic stress field near the root of a notch subject to tensile loading can be written approximately for theT = 0 direction as follows : K § U · .¨1  ¸ , 2Sx © 2 x ¹

V yy

(4.18)

K

U · § .¨1  ¸ . 2 x¹ 2Sx ©

V xx

(4.19)

The origin of the co-ordinate system xyz is located at the distance x= (U/2) +r. In other words, the stress field ahead of the notch root is similar to that of a crack with the tip at point O where the stress intensity factor is equal to K. At the notch root the maximum stress is equal to:

V max V yy§¨ x ©

U · ¸ 2¹

K 2U

.

(4.20)

Vyy (notch) Vyy (crack) Vyy,max crack singularity

Notch

U

O

Distance

Crack U/2 Figure 4.9: Elastic stress distribution of the normal stress Vyy according to Creager and Paris [4.7].

The elastic stress concentration is defined as

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kt

V max Vg

2K

SU

.

(4.21]

and the notch stress intensity factor as KU

10

k t V g SU .

(4.22)

log (Vyy/VJ) Pseudo stress singularity zone

Vyy,max

5 0.5 1 Log (U/2B Notch tip 1 10-4

10-2

log(r/B)

1

Figure 4.10: Distribution of the stress normal to the crack plane divided by the net stress (presented in bilogarithmic axes) according to Creager and Paris [4.7], B is the thickness.

For a notch with an infinite acuity the notch stress intensity factor is given by the limit of the expression (4.23) when the radius tends to zero: K *I

lim § r  o 0. V max yy ¨© r 2

S

U ;T 2

0 ·¸. U . ¹

(4.23)

Figure 4.10 presents in bi-logarithmic axes the distribution of the stress acting normally to crack plane divided by the net stress. We can see that the stress singularity has disappeared and we have a finite maximum stress at the notch tip. However, at a short distance from the notch tip the stress distribution is similar to that of a crack. We can speak in terms of a pseudo-stress singularity in this region. This pseudo-stress singularity is governed precisely by the notch stress intensity factor KU and has a1/—r dependence.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

77

4.3.2 NIU ET AL SOLUTION [4.8] In this case the elastic or the elastoplastic stress distribution is computed by using a finite element method. It has been proposed that the stress distribution is presented using bi-logarithmic axes as in figures 4.11.a and 4.11.b, where the logarithm of the non dimensional normal stress is plotted versus the logarithm of the non dimensional distance r/b where b is the ligament size. The elastic stress distribution diagram (figure 4.11.a) can be divided into 3 zones: í zone I where the non-dimensional normal stress is practically constant and equal to the stress concentration factor Vyy/Vg = kV. The maximum stress in this case is at the notch tip. í zone III where the non-dimensional normal stress exhibits a power dependence with the non dimensional distance,( C andD are constants with a slope Dd 

V yy Vg

C



r D . b

(4.24)

í zone II intermediate between zone I and zone III. The elasto plastic stress distribution diagram (figure 4.11.a) can be divided in to 4 zones : í zone I where the non dimensional normal stress increase to its maximum value. í zone III where the non dimensional normal stress exhibits a similar power dependence with the non dimensional distance .This zone is generally shorter than in the elastic case and the exponent Dis generally greater than2 í zone II intermediate between zone I and zone III. í zone IV where the non dimensional stress tends to 1In zone III, for the elastic or elastoplastic cases, the distribution of the stress normal to the notch plane can be expressed by the following relationship for x ” Xef: V yy

,

KU

 2Sr

(4.25)

D

where KU is the so-called notch stress intensity factor. Xef is the socalled effective distance which corresponds to the beginning of zone III or the pseudo-singularity zone. It corresponds to the end of the highly stressed zone at the notch tip which plays an important role in the fracture process. Along the pseudo-singularity line the product of the stress and the distance to the power D is constant. For the particular data pair [Xef, Vyy, (r = Xef)], we can get the value of the notch stress intensity factor.

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KU



V yy x



X ef . 2S X ef

D .

(4.26)

log (Vyy/Vg)

Log (Vyy/Vg) Vmax

D 1 D 1

Log (r/B) Figure 4.11.a: Non-dimensional elastic stress distribution at the notch tip versus the non dimensional distance in a bi-logarithmic graph. (3 point bending specimen).

log (r/B) Figure 4.11.b: Non-dimensional elastoplastic stress distribution at the notch tip versus the non dimensional distance in a bi-logarithmic graph. (3 point bend specimen).

4.4 Global stress criterion for fracture emanating from notches.

Using the above mentioned notch stress distribution, two fracture stress criteria (namely global and a local) may be derived. To establish a global stress criterion, the following two points may be considered: - for the critical events, the notch tip stress distribution characterised by the notch stress intensity factor reaches a critical position: c , (4.27)

KU KU

KcUis the critical notch stress intensity factor. - This critical notch stress intensity factor is related to the critical load Pc in a formula given by: Pc . F U a W ,\ , (4.28) K cU B.W 1  D P

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

79

where B is the thickness, W the width of the specimen and F UP a geometrical correction. The parameter K cU is a measure of the fracture toughness having the following units. MPa.mD. 4.4.1 CARPINTERI’S SOLUTION [4.9] In literature [4.9], relationships between the applied gross stress or load and the stress intensity factor for a specimen having a notch of infinite acuity can be found. Solutions can also be found for a plate in tension with a notch of infinite acuity and for a three point bend specimen with a notch of infinite acuity, see figures 4.12 a and 4.12.b. For the first case: (4.29) K *I \ V g .W D . F V a W .



Using the similitude principle from the Buckingham theory, when \ o S, D o 0, we obtain: 1 . (4.30) FV aW 1 a W



For a three point bend specimen with a notch of infinite acuity: PL . F V a ,\ , K *I \ W BW 2  D





(4.31)

where L is the span, B the thickness and W the width of the specimen. The geometrical correction function can be expressed as the product of two functions c(\) and g(a/W) of separate parameters\ and a/W . (4.32) F V a W ,\ # C \ .g a W ,







Function c(\) exhibits two extreme values c(0) = 0.5 and c (S) = 1 and can be approximated by the following relationship (where E is an unknown exponent) : C \ #



1 ª \ Eº .«1  ». 2« S »¼ ¬

(4.33)

4.4.2 VERREMAN’S SOLUTION [4.10] In order to determine the stress distribution in the near region of a weld toe, Verreman has used the Williams’s solution [4.2]. He considered that the stress distribution is governed by the stress intensity factor KI* having an exponent Dcorresponding to the Williams’s

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solution for an open notch angle < at the weld toe. The stress normal to the notch plane divided by the gross stress is given by:

V yy Vg

O 0.



a D , B

(4.34)

where O0 is a parameter . More generally this distribution is given by :

V yy Vg

a D 4 a i . ¦ . B i 0 B





(4.35)

P

< a W

W

Vg

Vg

a

< L



K *I \ V g .W D . F V a W

(4.36)

Figure 4.12 a :notch stress intensity factor for a plate in tension with a notch of infinite acuity (U= 0).

K *I \

PL . F V a ,\ W BW 2  D



(4.37)

Figure 4.12 b : notch stress intensity factor for three point bend specimen with a notch of infinite acuity (U= 0).

Equation (4.34) reduces to equation (4.35) when a/B is small. Considering the stress intensity factor to be given in (4.38) : (4.38) K *I M K V g Sa .

Mk is a correction factor which is given by: MK

a D 4 a i . ¦ Oi li , B B i 0





(4.39)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

2 i zi 1 ³ S 0 1 2 z

Oi

81

*§¨ i  D  1 ·¸ 2 2¹ . © . (4.40) S *§ i  D  1· ¨ ¸ 2 ¹ © 2

For a small notch length MK can be found from (4.41): a D , (4.41) M K O 0 l 0. B In addition, Verreman proposed that the stress distribution attains the gross stress value upon a distance proportional to BD. The value of K,* is then given by:



KI

V g H 0 B D . 2S .

(4.42)

Log (Vyy/Vg) 6

<

4 D

2

1 H0 BD (2S)0.5

1 0.6 10-5

10-4

10-3

10-2

1 10 Log (r/B)

Figure 4.13: Distribution of the stress normal to the crack plane divided by the gross stress presented in bilogarithmic axes according to Verreman [4.8]. B is the thickness.

4.4.3 APPLICATION TO THE FRACTURE TOUGHNESS OF GLASS Using three point bend specimens made of glass ‘Float’, fracture toughness determination has been carried out. Specimens with identical notch length but different notch radii (U = 0.125, 0.4, 0.5, 0.6 and 0.75 mm) were used. In this case, notch specimens are preferred to precracked specimens because it is very difficult to precrack very brittle materials such as ceramics and high strength steels. Until now, fracture

82

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toughness determination has been based upon the ASTM standard for precracked specimens. Alternatively, the approach adopted by Creager [4.7].for the notch stress intensity factor is equal to : K *Ic

S 2

lim r  o 0 V yy r

U 2 . r .

(4.43)

When we transform the notch to that of a crack, application of the ASTM’s standard formula leads to the following values of KIc. K Ic



Pc .F P a W B W

(4.44)

Pc is the critical load and FP(a/W) a geometrical correction factor.

1.5

Fracture toughness (MPa¥m) K*Ic KUc KIc

1

¥(Notch radius) (¥mm) 0.5

0

05

1

Figure 4.14: Influence of the square root of the notch radius on fracture toughness parameters KIc, K*Ic and KUc.

The fracture toughness parameters KIc, K*Ic and KUc have been determined from experimental results and are reported in figure 4.14 as a function of the square root of the notch, where in this case the notch angle is equal to zero the exponent Dis equal to 0.5. It can be seen that the fracture toughness values are very similar and increase as a function of the square root of the notch radius.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

83

4.5 Local stress criterion for fracture emanating from notches It is assumed that the fracture process requires a physical volume. This assumption is supported by fracture resistance being affected by loading mode, structure geometry and scale effects. The value of the’hot spot stress’ being the maximum stress value, is unable to explain the influence of these parameters on the fracture resistance. We have to take into account the stress value and the stress gradient at all neighbouring points within the fracture process volume. This volume is assumed to be quasicylindrical by analogy with a notch plastic zone of similar shape. The diameter of this cylinder is called the ‘effective distance’. By computing the average value of stress within this zone, the fracture stress can be estimated. This leads to a local fracture stress criterion based on two parameters: the effective distance Xef and the effective stress Vef. A graphical representation of this local fracture stress criterion is provided in Figure 4.15, where the stress normal to the notch plane is plotted as a function of the distance ahead of the notch. For a determination of Xef, a graphical procedure is used. It has been observed that the effective distance is related to the minimum value of the relative stress gradient Fas defined by : 1 d V yy F . . (4.45) V yy dr log Vyy High stressed zone

Pseudo Stress Singularity zone Log Vef

Notch

log r Fracture process zone log Xef Figure 4.15: Schematic presentation of a local stress criterion for fracture emanating from notches.

It can be easily shown that this distance corresponds to the beginning of the pseudo stress singularity. Definition of the effective distance as the distance of minimum relative stress gradient is indicated in this figure. Charpy V notch specimens made in a CrMoV steel (yield stress 771 MPa) were tested statically in bending at one selected (lower shelf region) temperature. The tensile stress

G PLUVINAGE

84

distribution at the notch has been calculated using a FEM. A 2D model under plane strain conditions was used for the elastic-plastic analysis. The effective distance Xef has been determined using normal stress distributions below the notch root plotted, in bi-logarithmic axes. The relative stress gradient (equation 4.45) has been plotted on the same graph for the purpose of obtaining a precise value of the effective distance. For a fracture load equal to 131 KN, the effective distance has been found to equal 0,380 mm. The effective stress is defined as the average of the weighted stress inside the fracture process zone

V ef

1

X ef

³ V ijdx . X ef 0

(4.46)

For this material the mean value of the effective stress is 1223 MPa which can be compare to the average maximum local stress at fracture, Vmax of 1310 MPa.

3.5

Logarithm normal stress (log Vyy) Vyy

3.0 2.5 2.0

F CrMoV rotor steel Critical load Pc =13,1KN

log Xef

1.5 1.0 -1.2

-0.8

-0.4

0.0 0.4 logarithm distance (log r)

Figure 4.16: Notch root stress distribution at notch root together with the relative stress gradient versus distance from the notch tip for a fine carbide CrMoV rotor steel.

4.5 Notch effects on the brittle-ductile transition Some materials, for example ferritic steels, exhibit a brittle to ductile transition. This transition is promoted by increasing the temperature or decreasing the loading rate. This effect can be explained by the fact that plasticity is necessary to initiate the fracture process and plasticity is a thermal activated process. A third physical parameter influences this transition, notably the notch radius as controlled via the level of stress triaxiality. Based on different experimental observations of KIc versus —U and for a

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

85

critical notch radius Uc,it is noted that KIc is independent of U. Similarly, the presence of a ductile plateau is independent of the notch radius. In the transition regime, KIc is a linear function of the square root of the notch radius and a similar transition curve can be drawn which represents more generally, the notch effect on fracture toughness. However,in order to take into account for any notch geometry, the critical notch stress intensity factor is plotted versus U D figure 4.17. The notch sensitivity m can be represented by the slope of the transition line [4.11].

m

tan g E .

(4.47)

Critical notch stress intensity factor KcU

Brittle

Transition

Ductile

Temperature = cst Strain rate = cst Critical notch radius Ucr (Notch radius )D Figure 4.17: Evolution of the critical notch stress intensity factor KUc versus the notch radius at the power D. Definition of the critical notch radius.

As the temperature decreases, the material becomes more and more brittle and the notch sensitivity increases, i.e. notches are more dangerous at low temperature (Figure 4.18). The ductile-brittle transition is sensitive to the applied loading rate, as shown in figure 4.19. With increasing loading rate, the material’s yield strength increases along with a corresponding increase in brittleness figure 4.19. Consequently its notch sensitivity increases.

86

G PLUVINAGE

 

             

Critical notch stress intensity factor KcU T1>T2>T3

T1 T2 T3 STRAIN RATE = Cst (Notch radius )D

Figure 4.18 : Evolution of the critical notch stress intensity factor versus UD for three different temperatures

Critical notch stress intensity factor KcU

Static Dynamic TEMPERATURE = CST E1

E2

(Notch radius)D

Figure 4.19: Evolution of the critical notch stress intensity factor versus U Dfor two loading rate conditions. 

4.6 Notch sensitivity in mixed mode fracture

Notch sensitivity has been measured using ring specimens made of high strength steel (French standard 45 SCD 16). A description of the specimen is given in figure 4.20. The ring has a small notch of 4 mm depth and variable notch radius in range [0.2 – 2 mm]. The notch plane is inclined from the loading direction at 3 different angles in order to promote different modes of loading. mode I E = 0° mixed mode I + II 0° < E < 33° mode II E = 33° After heat treatment, material has the following mechanical properties.

Yield stress Ultimate strength Elongation KIC 1 463 MPa 1 162 MPa 2,8 % 97 MPa—m Table 4.2 mechanical properties of 45 SCD 16 steel.

It has been observed that the critical load increases with inclined angle and also with notch radius, when greater than a critical value Uc = 0,75 mm.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

y

87

A

F A x

F

7mm

U

I 20 mm I 40mm

Figure 4.20: Geometry of a ring specimen used for mixed mode I +II fracture.

Circumferential stress has been computed by a finite element method and the effective distance has been determined using a graphical procedure involving the minimum value of the relative stress gradient. The effective stress has been computed by averaging the circumferential stresses over the effective distance. According to Erdogan and Sih’s criterion [4.12] for mixed mode of fracture, the effective stress can be expressed versus value of mode I and mode II notch stress intensity factor V ef

1



2S V ef

D



Tª T 3 º, cos « K IU cos 2  K IIU sin T » 2 2 2¬ ¼

(4.48)

where T is the crack bifurcation angle, KIU mode I notch stress intensity factor and KIIU mode II notch stress intensity factor. Fracture toughness has been determined using a local notch fracture criterion involving effective stress and effective distance. It has been noted that effective distance is related to the notch radius through the following relationship. (4.49) Xèf = A’.Ua’. with a’(E) and A’(E) parameters which are dependant of inclined angle. Similarly, the effective stress is also related to the notch radius and loading angle by: Vcef= B’/Ub’. (4.50) with B’(E) and b’(E) parameters dependent of the inclined angle. Fracture toughness as determined from the equivalent notch stress intensity factor has been plotted versus the notch radius. We can note that the equivalent critical notch stress intensity factor is practically constant up to a critical notch radius of Uc # 0.70.

88

G PLUVINAGE

K II K Ic

A

 KI K Ic

2 B

KI C , K Ic

(4.51)

A new criterion for mixed mode brittle fracture in notched specimens is based on the criterion proposed by Erdogan and Sih [4.12]. For this criterion, fracture of cracked specimens occurs when the product of the critical circumferential stress and square root of the distance reaches a critical value c 2Sr K Ic , (4.52) V TT FV where KIc is the fracture toughness and FV a geometrical factor. For notched specimen, the criterion is modified as follow c 2Sr V TT , ef 

D

K cU , ef ,

(4.53)

where D is the pseudo singularity exponent. This formula uses the critical effective circumferential stress and the effective critical notch stress intensity factor. This criterion can be written as : cos cos

T 2

>K U , I .sin T  K U , II 3 cosT  1 @

Tª

T 2 3



« K U , I . cos 2 «¬ 2



º K U , II sin T » 2 »¼

0, KC U, I .

(4.54) (4.55)

The assumptions generally used for steels are that KIc = KIIc, consequently:

K cU , I

K cU , II .

The fracture criterion can be rewritten as: ª º K U , II T K U, I  cos « . sin T  3 cos T  1 » 0 , » 2 «KC KC U , II «¬ U , I »¼ ª

cos

T « K U, I 2 «KC ¬« U , I



. cos

º sin T » 1  » 2 KC U, I ¼»

T 2 3 K U , II 2

K U, I KC U, I

§ · ¨ K U , II ¸ f¨ ¸ ¸ ¨ KC © U, I ¹

(4.56)

(4.57)

(4.58.a)

(4.58.b)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

Equation (4.58. b) can be plotted in the plane: 2 § · K U , II K U, I ¨ K U, I ¸  B¨ A ¸C , ¨ K cU , I ¸ K cU , I K cU , I © ¹

§ ¨ ©

· ¸ ¹

89

(4.59)

with A = -0.247 U - 0.6 ; B = 0.35 U - 0.1 ; C = 0.1 U + 0.86 This curve is called the intrinsic fracture curve and it can be noted that : K U, I

K cU , I

d 0.6 .

(4.60)

There is no influence of notch radius on this intrinsic curve. In mode II and for a crack it can be seen: (4.61) K cU , II 0.86 K cU , I . a value given by the Erdogan and Sih criterion. 1.0

c

KU KU , II

U 0.15 mm

,I

U= 0.30 mm U= 0.50 mm U = 1.00 mm U = 2.00 mm crack

0.8 0.4

KU KU

0

, II

0.0

0.2

,I

! 1.2

KU KU

0.4

0.6

, II

,I

 1.2

0.8

1.0

c

KU KU ,I

,I

Figure 4.21: Intrinsic failure curve for mixed mode I+II fracture on Steel 45 SCD 6.

Notch sensitivity appears only, for the range of explored notch radius and for this steel only if the bimodality ratio is more than 1.2. K U , II K U, I

! 1.2 .

(4.62)

4.7 Conclusion

Table 4.3 below provides a summary of the definitions of the stress and the notch stress intensity factor. For a notch, the fracture criterion is expressed as the critical Notch Stress Intensity Factor with the following unit of MPa.mDThis criterion can be considered as an ‘associated parameter’ fracture criterion. Dissociation of these parameters is introduced in a local stress fracture criterion where consideration is given to the effective distance

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90

and stress. An alternative is to consider the average stress which acts in the process zone of size Xef. Plate with a crack

Notched plate with infinite acuity (zero notch radius)

P

P

P

Vyy

V yy o

r

notched plate with a notch radiusU

\

V yy o

\

r

o

r

a P

P

KI for a crack K I = lim 2Sr V y y ro0

P

K*I for infinitely sharp notch

K , = lim 2S ro0

D

V yy r

D

KUfor a notch with a notch radiusU KU V yy Sr

D

Table 4.3: Definitions of the stress intensity factor and the notch stress intensity factor.

REFERENCES [4.1] Irwin. G.R (1948).’Analysis of stresses and strain near the end of a crack traversing a plate’ Trans ASME Journal of Applied Mechanics, 24, pp 361-364, [4.2] Williams. M.L. (1952 ). ‘ Stress singularity resulting from various boundary conditions in angular corners of plates in extension,’ Journal of Applied Mechanics, Vol.19, N°4, pp 526~528. [4.3] Gross. B. and Mendelson. A, (1991). ‘Plane elastoplastic analysis of V notched plates4, International Journal of Fracture. Vol. 48, pp. 79-83. [4.4] Hasebe.N. and Kuntada.Y, (1978). ‘Calculation of stress intensity factor from stress concentration factor’, Engineering Fracture Mechanics, Vol. 10, N0. 2, pp. 215~221. [4.5] Lin .K. Y.and Pin. Tong. (1995). ‘ Singular finite elements for the V notched plate ‘ , International Journal for Numerical Methods in Engineering, Vol. 15, pp. 503~512. [4.6] Knésl. Z. (1991).’Criterion of V notched stability’, International Journal of Fracture. Vol. 48 , pp.79~83 [4.7] Creager. M. and Paris. P. C. 1967).’Elastic field equations for blunt cracks with reference to stress corrosion cracking’, International Journal of Fracture. Vol. 3, pp. 247~252. [4.8] Niu. L .S , Chehimi .C and Pluvinage. G. (1994). ‘Stress field near a large blunted V notch and application of the concept of critical notch stress intensity factor to the fracture of very brittle materials’.Engineering Fracture Mechanics ; vol 49, n°3, , pp 325~335. [4.9] Carpenteri. A. (1995). ‘Stress singularity and generalised fracture toughness at the vertex of re-entrant corners’, Engineering Fracture Mechanics, Vol. 56, N° 1, pp. 143~155. [4.10] Verreman.Y, Dickson J.I. and Bailon.J.P. (1989). ‘Generalisation of the Kitagawa diagram to V-notch members’, Advances in Fatigue Science and Technology, Kluwer Academic Publishers, pp 785~.798. [4.11] Toth L. (1995). Proceedings of 1st ILSSCRSS Meeting Miskolc. [4.12] Erdogan.F. and Sih.G.C. (1963)‘On crack extension in plates under plane loading and transverse shear’. Journal of Basic Engineering.

CHAPTER 5 ENERGY CRITERIA FOR FRACTURE EMANATING FROM NOTCH ______________________________________________________________________ 5.1 Introduction Fracture emanating from notches can be predicted by energy criteria. Two ways are possible: we can consider that the fracture energy is provided by the elastic or inelastic energy stored by the structure or we consider that it is derived from the fracture process volume. In the first case we can adopt a global fracture criterion, in the second case a local fracture criterion. The fracture energy is defined as the difference between the external work and the stored energy per increment of new fracture surface. It corresponds to the variation of potential energy d3. Per unit thickness, given ‘da’ as the length of the newly created fracture surface, we can write : ~ d3 dU  G or G . (5.1) da da G is used in elastic case and is called the strain energy release rate. In case of non-linear ~ elasticity the non-linear strain energy release rate G is used, U is the work done. Rice et al [5.1] has demonstrated that in the elastoplastic case, this potential energy variation can be described by a path independent integral called J: wu J ³ W * dy  T i i dS , (5.2) wx * where the strain energy density W* is defined as : W * ³ V ijd H ij ,

(5.3)

where Vij and Hij are respectively values of stress and strain in their respective tensors, Ti are tensions on surface, ui the displacements and S a contour. At fracture, the J integral reaches a critical value JIC which represents the fracture toughness of the material. Turner [5.2] has postulated that this fracture toughness is proportional to the work done for fracture Uc. Uc , (5.4) J Ic K . Bb

where K is a factor of proportionality, % the thickness of the specimen and b the size of the ligament. The influence of the notch acuity on the J integral has been studied by different authors, e.g. Firao [5.5] and a linear relationship has been found. However the influence of the 91

92

G PLUVINAGE

notch radius on fracture toughness value is seen through its effects on both the values of eta factor and the work done for fracture. Adopting the local criterion for fracture emanating from notches, Czoboly et al [5.4] first introduced the use of strain energy density W*. The local fracture criterion assumes that at a critical event, the strain energy density reaches in the fracture process volume, a critical value W*C. Czoboly et al found that this critical value decreases exponentially with the stress concentration factor. 5.2 Influence of the notch radius on J integral

In Chapter 3, the path independent integral M was defined by: ª § wu j ·º M ³ «W * . x k  V jk ¨ x l  2N u j ¸» n k ds , ¨ wx ¸» l © ¹¼ *¬«

(5.5)

i, j, k, l = x, y ; W* is the strain energy density, xk is a co-ordinate, the product Vik.nk correspond to the traction on the path * and uj is the displacement. We also define a coefficient N which can be can be determined using the stress strain diagram (figure 3.5). By line integral transformation, we have the following expression for the M integral : § wu j ·º w ª «W * . x k  V jk ¨ x l  2N u j ¸»dxdy . (5.6) M ³ ¨ wx ¸» l © ¹¼ * w x k ¬« For a notch with y = 0, the M integral reduces to: SU V 2max M , 4E

(5.7)

which leads to a linear relationship between the M integral and the notch radius. Firao [5.5] has described the strain distribution at the notch tip by the following relationship : ª n  0.5  1 . n  1.5 .*n  0.5 º , ». *0.5 .*n  1 ¬ ¼ Re H y U

H yy H y . «

(5.8)

where n is the strain hardening exponent, I the Shi’s constant and * the gamma function defined by : t * ³ t n  1. et .dt . (5.9) 0 At the critical event

H yy H cl and U

Uc .

(5.10)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

93

where F [*(n)] is defined as: ª n  0.5  . n  1.5 .*n  0.5 º F >*n @ « » , (5.11) *0.5 .*n  1 ¬ ¼ 1 n ­ ½ .H y ° ° H c  n 1 (5.12) J Ic Re ® ¾. F >*n @ ° ° ¯ ¿

JIc is equal to :

Firao has verified this relationship for 3 steels with different sulphur contents finding a linear relationship between JIC and the notch radius U above a critical value, see Figure 5.1. Fracture toughness (KJ/m2)

500

Steel 1 S% = 0,007 Steel 2 S% = 0,017 Steel 3 S% = 0,034

400 300 200 100 Notch radius (mm) 0

0.0

0.1

0.2

0.3

Figure 5.1: Influence of notch radius on fracture toughness according to [5.3].

5.3 Influence of the notch radius on the eta coefficients. 5.3.1 DEFINITIONS OF THE COEFFICIENTS K, Kel, Kpl The definition of the eta coefficient can be given by the following formula: J

K

U Bb

.

(5.13)

This definition is valid for a linear elastic or non-linear behaviour. For an elasto-plastic behaviour, Sumpter [5.7] has defined the Kel and Kpl coefficients as follows :

94

G PLUVINAGE

J

K el

U pl U el  K pl , Bb Bb

(5.14)

where Uel and Upl are the elastic and plastic components respectively of the work done for fracture. 5.3.2 ANALYTICAL SOLUTIONS FOR K, Kel, Kpl, Kcan be written as follows : b wU K  U wa

J 

1 wU B wW  a

W  a U



W  a wU U

wa

,

wU wLnU . wW  a wLnW  a

(5.15)

(5.16)

5.3.2.1 Determination of K, Kel J el

G

P 2 wC , 2 B wa

(5.17)

where C is the compliance of the specimen. K el

P 2 wC Bb , 2 B wa U el

(5.18)

K el

b wC , C wa

(5.19)

In the case of elastic behaviour, the K factor is determined from the change of compliance with crack length.

5.3.2.2 Determination of Kpl An analytical solution for can be found in the following case (5.8): í pure bending specimen, í non symmetric tensile specimen, í CT specimen. x pure bending specimen (Figure 5.2) In this case, we make the assumption that the strain is linearly distributed on each side of the neutral axis. The maximal strain Hmax is equal to: W  a , (5.20) H max O 2 

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

O is a constant. The elongation S is given by: 2U S ³ H maxdy H max 2 U , 0

(5.21)

where U is the notch radius and the bending T angle is equal to : 4 U H max 4 UO W  a S 2 UO . T W  a) 2 W  a W  a

(5.22)

In this case O is given by:

T

O

2U

.

(5.23)

Furthermore it is assumed that the material behaviour obeys to Ludwik’s law :  V = V*. Hn. (5.24)

M Rigid

(A) Plastic a

H

W (B) V = V*.Hn

H = Ox

x

(W-a)/2 Figure 5.2: (a) Model for plastic flow of a notched beam subjected to a bending moment M. (b) Schematic of the strain distribution on the ligament.

95

96

G PLUVINAGE

The bending moment M is equal to : (W  a ) / 2 (W  a) / 2 V * H n xdx , M 2 ³ Vxdx 2 ³ 0 0 (W  a) / 2 M 2V * O n x1  n dx . ³ 0

M

2V * O n . 2n

§  W  a · ¸ ¨ © 2 ¹

2n

(5.25)

2V * W  a

2n

2  n . 2 2  n

and the work done for fracture U is equal to : 2  n 1 n V * W  a .T U ³ MdT , n 1  n  2 U 2  n . 22  n

wU wa

V * W  a

 2U

n

2n

.T n  1

2  n . 21  n

.

.



T n , 2U

(5.26)

(5.27)

.

(5.28)

In this case K is given by =

K pl



W  a . wU U

wa ,

Kpl = 2+n.

(5.29) (5.30)

non symmetric tensile loading specimen (figure 5.3) The total elongation 't can be divided into tensile 'p and bending 'm elongations. (5.31) 't ' p  ' m 'p

H 2U ,

(5.32)

where H is the average strain in the plastic zone. 'm

S

W a , W a

(5.33)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

S

(5.34)

H max 2 U

(W-a)/2

a

2U

(W-a)/2

plastic

U

B

Rigid W

Figure 5.3: Model of plastic flow for non symmetric tensile loading specimen.

Again the material behaviour is assumed to obey Ludwik’s law. 1 V n. H V*



(5.35)

The maximum strain is equal to :

H max

1

 V max V*

n

,

(5.36)

and the total elongation is given by : 't

W  a º , 2U ª 1 n V n  V 1max « 1 W  a »¼ V* n¬





1 n º ª « § W  a · 3 W  a » .«1  ¨ ¸. ». 't n 1 « © W  a ¹  W  a » V *1 n . B1 n  W  a «¬ »¼ The stress (V and the maximum stress (Vmax) are given by: 2 U1 n

§ ¨ ©

· ¸ ¹

(5.37)

(5.38)

97

98

G PLUVINAGE

V

P , B.W  a

V max

PW  a .V 2.B. W  a

2

,

(5.39)

where the load (P) is given by : 1 V *n B.W  a . , P ' tn . n 1 nº ª W  a  2U «1  31 n . » W a »¼ ¬«

(5.40)

and the work (U)done is given by : 1 V *n B.W  a . U Pd , ' tn  1. t n 1 nº ª  W a n  U  2 . 1 n 1  «1  3 . » W a «¬ »¼

(5.41)



³ '



½ ­ ° ° n °   . B W a W a  w  wU °° ° V* . ® ' tn  1. ¾ n 1 nº wa ª wa  2 U .n  1 °° «1  31 n . W  a » °° W a »¼ °¿ °¯ «¬ wU V *n B.W  a . 1 . ' tn  1. 2n n wa  2 U .n  1



(5.42)

A

­ ª º½ °° n n 1« 1 n n  1 W  a 1 n » °° 2W n . . ® 1 A   W  a A «3 . » ¾ (5.43)  2 n W a ° «  W a  »¼ °°¿ °¯ ¬ 1 A with ª W  a 1 nº «1  31 n . » W a «¬ »¼ V *n B.W  a .  1 .ª«1  2. 1 n n  1 .§ W ·. 1 . W  a 1 n º» . (5.44) ¨ ¸ 3 ' tn  1. 2n « n W a¹ A W a © »¼ A ¬  2 U .n  1







The value of Kpl can be found from:

K pl



W  a . wU U

wa

.

(5.45)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

99

º ª · § ¸ ¨ » « n 1 n 1 § W ·¨ ¸ W a » . (5.46) .«1  2. 31 n n  1 .¨ . ¸.¨ » « n 1 n ¸ W  a ©W  a ¹¨ W a  ¸ » « ¨ 1  31 n. ¸ »¼ «¬ W a ¹ ©





Example for (w+a)/(w-a) =3, the relationship between n and Kpl is given in table 5.3.

n Kpl

0 2,33

0,1 2,46

0,25 2,66

0,5 2,99

1 3,57

Table 5.3 K pl values for different n values.

For a particular valueKpl = 2, U/w = 0.0545. tension and bending (figure 5.4) A model of the plastic flow for a specimen subjected to tension and bending is given in figure 5.4. In this case, the elongation is: 2 2 L.P 2 U .P 2 LP W 2 P. W  a    , (5.47) 't EF E Fn 4 EI 4E I n B W  a 3 BW 3 ,In . where F BW , F n B.W  a , I 12 12 For this we find : § U 2P ª 4L W  a 2 ·¸º» .« .¨1  3  ; 't EB « W W  a ¨ W  a ¸¹» © ¬ ¼



Substituting

W a W a

C,

.

(5.48)

4L U §¨ W  a 2 ·¸  . 1 3 W W  a ¨ W  a ¸¹ ©



F,

(5.49)

We find the load P is given by P

EB ' . 2F t

(5.50)

100

G PLUVINAGE

Work done for fracture is equal to :

U

wU wa

wU wa

EB ' t2

³ Pd ' t

EB ' t2 4

4F wF . wa F2

(5.51)

,

(5.52)

,

12CW  1  3C 2 EB ' t2 W  a . 4 W  a 2



,

(5.53)

Figure 5.4: Model of plastic flow for specimen subjected to tension and bending.

W  a . wU

, U wa 12CW  1  3 C 2 .W  a  W  a U. . W  a 2 .F

K pl







(5.54)

(5.55)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

Example CT specimen (figure 5.5) For the above case a = 0,5 W ; H = 1,2 ; 2L =1,2 W ; a = B = 0,5 W and C

W a W a

1.5W 0.5W

F= 2,4 + 56 (U/W).

3

U

K pl

W

200

.

2,4  56

U

.

(5.56)

W

(Kpl = 2 for U/w = 0,0545)

P

(W+a)/2 Rigid 2U

Plastic

'

S

'M 'p a

(W-a)/2

(W-a)/2

W

Figure 5.5: Plastic flow for a CT specimen.

101

102

G PLUVINAGE

Pure tension (figure 5.6) The value of the applied stress (V) is given by :

V

P , B.W  a

V *H n

(5.57)

where the elongation 't 't

Applying Ludwik’s law:

H

2U ³ Hdy 0 1

 V V*

(5.58) n,

(5.59)

W/2

2U

W/2

Plastic

U

Rigid

B

W

Figure 5.6 : Plastic flow for a specimen subjected to pure tension.

We obtain : 1

't



't



P

V * BW  a

V V*

n .2 U ,

P V * BW  a

 2U

n

(5.60) 1



' nt .

n

.2 U ,

(5.61)

(5.62)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

Here the work done is given by: U

³ Pd ' t

V * BW  a n

 2 U 1  n

§ · ¨ V * B '1  n ¸ ¸, ¨ ¨¨ 2 U n 1  n ¸¸ © ¹

wU wa

KU

J

'1t  n ,



B W  a

(5.63)

(5.64)

1 wU B wa ,

(5.65)

1

(5.66)

We find the Kpl factor: n

K pl

W  a . 2 U .1  n §¨ V * B. '1  n ·¸ ¸ .¨ V * B.W  a . '1  n ¨¨ 2 U n .1  n ¸¸  ©

¹

This leads to particular value whereKpl = 1

2.4

Kel factor

2.2 2.0

Deep crack solution

1.8

U= 2mm U= 1mm U= 0.25 mm U= 0 (crack)

1.6 1.4 0.2

0.3

0.4

a/W

0.6 0.8 0.7 0.5 Non dimensional crack length

Figure 5.7: Values of Kel for notches and cracks as the function of the ratio a/W for elastic behaviour.

103

104

5.3.3

G PLUVINAGE

INFLUENCE OF NOTCH RADIUS ON K EL

In order to determine the influence of notch radius on Kel, a numerical study was carried out under plane strain conditions. Figure 5.7 presents a plot showing the evolution of Kel versus the relative notch depth for constant notch radius value. For cracks we notice a good agreement with results in the literature and values are close to the value of 2 given by Rice et al [5.1]. Comparison between the values obtained for different notches and cracks having the same length reveals that the K notch / K crack ratio remains constant for a given acuity. el el Furthermore, the difference between the coefficient Kel for notches and cracks, of the same length, does not exceed 10% of the later. 2.2

Kel factor

2.0 1.8

a/W = 0.25 a/W = 0.45 a/W = 0.65

1.6 1.4 0

0.5

1

1.5

2

Notch radius (mm) Figure 5.8: Values of Kel factor versus notch radius for notches and crack, (elastic behaviour). For deeper cracks, we notice that Kel factor increases with increase of notch radius and a/W (figure 5.8).

5.3.4

INFLUENCE OF NOTCH RADIUS ON THE K FACTOR

Figure 5.9 plots the evolution of K as a function of the relative notch depth a/W for constant notch radius U. Here elasto-plastic behaviour is assumed to follow Ludwick’s law. We notice that K increases with the relative depth a/W independently of U. Futhermore, the evolution of K as a function of notch root radius shows an absolute minimum denoted Uc in range [0.75 ~ 1 mm] (figure 5.10). Similarly, we notice that for radius values below Uc, K decreases linearly with an increase of U. Beyond this critical abscissa the K function increases withU and becomes approximately constant for radii ranging between 1.54 and 2 mm. The difference between the K determined for a crack and compared to that of a notch with the same length, can reach up to 36%. This is relatively important and justifies the present approach.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

105

5.3.5 INFLUENCE OF NOTCH RADIUS ON FRACTURE TOUGHNESS The material under study is a French standard "XC 38" steel having mechanical properties as reported in table 5.4. Elongation Ultimate Yield strength stress) A% Re ( MPa) Rm (MPa) 304

430

30

Ludwick law

Hardness Vickers Hv

Fracture toughness JIc KJ/m2

137

798

 V = 830 .Hn

Table 5.4 Mechanical properties of XC38 Steel.

A microanalysis of material gives the following chemical analysis in (Wt%) : C.0,24% 14% , Si, 0.53%,Mn 0,53 and traces of Cr, Ni, and Mo. The present experiments used U-notched specimens of dimensions 25 x 25 x 150 mm obtained from a 30 mm x 30mm plate, specimen length was parallel to the rolling direction.

K factor

3.0 2.5 2.0 1.5 1.0

U = 0 mm (crack)

0.5

U = 0.25 mm U = 1 mm U = 2 mm

0.0

0.2

0.3

0.5 0.6 0.7 0.4 Non dimensional notch depth (a/W)

Figure 5.9: Evolution of K factor as function of relative depth at constant notch radius.

Different notches were introduced employing a wire cut EDM (Electrical Discharge machine) using different diameter wires. Notch root radius was measured using a profile projector with a magnification x 50. The experimental displacements were measured along the gauge section of the specimen thereby avoid displacement contribution arising from the grips and columns of the testing machine. In these experiments, it is assumed that the critical non- linear fracture energy was reached at maximum load. This critical value of fracture energy is obtained as the area below the loadídisplacement curve from zero to maximum load Pmax.

106

G PLUVINAGE

K factor a/W = 0.45 a/W = 0.50 a/W = 0.55 a/W = 0.60

2.0

1.5

1.0 0

0.5

1

1.5 2 Notch radius U (mm)

Fig. 5.10: Evolution of K factor, as function of notch radius for constant relative depth.

The evolution of maximum load is plotted as a function of notch root radius at constant relative depth. Figure 5.11 shows that the critical loads increases linearly with increasing notch root radius UHowever, the energy absorbed until fracture increases non linearly with U.

80

Maximum load Pmax (KN) a/W = 0.2 a/W = 0.3 a/W = 0.4 a/W = 0.5 a/W = 0.6 a/W = 0.7

60 40

20 0 0.0

0.5

1.0

1.5

2.0

Notch radius Fig .5.11: Evolution of maximum load versus notch radius for different values of the ratio a/W.

The variation of fracture energy is plotted in figure 5.12 for different ratios of a/W. As can be seen from this figure, this relationship is non- linear.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

107

Non linear work done for fracture Ucnl (J) 800 a/W = 0.2 a/W = 0.2 a/W = 0.2 a/W = 0.2 a/W = 0.2 a/W = 0.2

600 400 200 0

0

0.5

2 1 1.5 Notch radius U (mm)

Figure 5.12: Evolution of the work done for fracture versus notch radius for different values of the ratio a/W.

Figure 5.13 presents values of the apparent fracture toughness determined using expression (5.67) for critical value of the work done U c a, U and associated value nl K(a,U). K a, U .U c a, U nl . (5.67) J Ic, Ap Bb

2.0

Apparent fracture toughness JIc,app (MJ/m2)

1.6

Numerical Experimental

1.2 0.8 0.4 0.0

Ucr = 0.85 mm

0.0

0.5

1.0

1.5 2.0 Notch radius U (mm)

Figure 5.13: Comparison between experimental JIc,App versus U and numerical results based on equation 5.67

From figure 5.13, it can be seen that for radii less than 0.85 mm, fracture toughness is independent of U. However for radii beyond 0.85 mm, JIc,App increases linearly with U

108

G PLUVINAGE

here JIc,App is given in (MJ/m2), U unit in (mm).Uc is termed the critical radius value and in this case is equal to 0.85 mm. This can be summarised as follows: * plateau: U < 0.85 mm,

JIc,App = Constant= JIC,

(5.68).

* linear evolution: U • 0.85 mm,

JIc,app = 8.6.10-4 U

(5.69).

Extrapolation of the linear portion of line intercepts the x,y axes at (0, 0). 5.4 Local energy criterion for fracture emanating from notches

5.4.1 STRAIN ENERGY DENSITY DISTRIBUTION AT THE NOTCH TIP A typical example of the strain energy density distribution plotted versus the distance from the notch tip (bilogarithmic axes) is given in figure 5.14. This distribution has been computed for axisymmetrical notched tensile specimen. The strain energy density distribution diagram can be divided into 3 zones: í zone I where the strain energy density is practically constant and equal to the maximum value: 2 W *max (5.70) k 2* , W W *N



í zone III where the strain energy density exhibits a power dependence with the non dimensional distance (C and D' are constants, D' > 1) : W* W *N

C

r D' , b



(5.71)

í zone II intermediate between zone I and zone III. The stress distribution in zone III can be assimilated to a “pseudo strain energy density singularity”. This distribution can be considered only for a distance greater than Xef defined on figure 5.14 with the following formula: K U,W * W* for r > X’ef. (5.72) D'  2S

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

109

Logarithm (strain energy density) ln W*

W*max

D’ W*N

1

Logarithm (distance); lnr Figure 5.14: Schema of strain energy density distribution at notch tip.

5.4.2 INFLUENCE OF THE NOTCH RADIUS ON THE STRAIN ENERGY DENSITY DISTRIBUTION The strain energy density distribution at the notch tip has been computed using finite element for axisymetrical specimen with different notch radii. The material is a steel having the mechanical properties listed in table 5.5. The stress strain behaviour of this material obeys Ludwik’s law.

Ultimate strength Yield stress strain hardening coefficient strain hardening exponent Young’s (Coulomb) modulus

Tension 1030 MPa  835 MPa 1313 MPa 0.075 212000 MPa

Torsion 580 MPa 879 MPa 0.064 82000 MPa

Table 5.5 mechanical properties of the used steel

The strain energy distribution is plotted in bi-logarithmic axes versus distance to the notch tip for the three modes of loading (tension, torsion and tension plus torsion). Each strain energy density distribution exhibits a maximum value W*max which depends of notch radius and loading mode. Figure 5.15 shows evolution of this maximum value W*max. We can notice that this value decreases when notch radius increases. The relative decreasing (ratio for sharp notch to blunt notch) is similar for the three modes of loading. The strain energy density distribution is also characterised byD', the slope of the curve in the pseudo-singularity zone. The evolution of this parameterD' versus notch radius is presented in figure 5.16. We can notice that this parameter increases with notch radius. Values in torsion are less than in tension and tension and torsion.

110

G PLUVINAGE

The beginning of the pseudo singularity corresponds to the abscissa of the effective distance Xef. We can see from figure 5.17 that the effective distance increases linearly with increasing notch radius. Maximum strain energy density W*max (MJ/m3)

7

Tension Tension + torsion Torsion

6 5 4 3 2 1 3

4

6

5

7

8

Notch radius U (mm) Figure 5.15: Evolution of maximum strain energy density versus notch radius at the notch tip for an axisymetrical steel specimen loaded in tension, torsion or tension + torsion.

0.0

Parameter D’ tension tension + torsion torsion

-0.5

-1.0

-1.5 3

4

5

6 7 8 Notch radius U (mm)

Figure 5.16 : Evolution of exponent D’ with notch radius in tension torsion or tension +torsion (axisymetrical steel specimen).

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

2.6

111

Effective distance (mm)

2.4 TENSION

2.2 2.0 1.8 1.6 1.4

3

4

5

6 8 7 Notch radius U (mm)

Figure 5.17 : Evolution of the effective distance with notch radius in tension (axisymetrical steel specimen).

5.4.3 Local energy criteria for fracture emanating from notches

It has previously been proposed that the fracture process requires a physical volume at the notch tip. This volume can be considered as cylindrical with a diameter equal to the effective distance. At fracture, the average strain energy density in the fracture process volume is equal to the critical strain energy density: W* = W*c.

(5.73)

Critical strain energy density W*c

(MJ/m2)

500 400 300 200 100 0 1

1.7

2.3

3

Elastic stress concentration factor kt Figure 5.18: Evolution of the critical strain energy density versus the elastic stress concentration factor (axisymetrical steel notched specimens).

Czoboly et al [5.4] found that this critical strain energy density decreases with the elastic stress concentration factor kt according to:

112

G PLUVINAGE

W *c

W *c,0 . k bt

(5.74)

where W*c,0 is the strain energy density for a smooth specimen and b is a constant. A typical example is given in figure 5.18.

5.5 CONCLUSION

A global and a local energy criterion can be used for fracture emanating from notches. Critical value of the energy parameter J gives the fracture toughness JIc. This parameter increases linearly with the notch radius beyond a critical value. The critical strain energy density defined as an average value in the fracture process zone can also be used as a fracture criterion. This value decreases as a power function of the elastic stress concentration factor.

REFERENCES 5.1. Rice. J. R, Paris. P. C. & Merkle J. G. (1973).‘Some further results on J Integral analysis and estimates’, ASTM - STP 536, pp. 231~245. 5.2 Turner.C.E. (1979).‘Methods for post yield fracture safety assessment’. Post-yield fracture mechanics pp. 23~210. 5.3 Liebowitz . H. and Eftis J. (1971). ‘On non linear effects in fracture mechanics’. Engineering Fracture Mechanics, pp. 267~281. 5.4 Czoboly.E. Havas.I. and Guillemot .F. (1982). ‘ The absorbed specific energy until fracture as a measure of the toughness of metal’ . Proceeding of Symposium on absorbed specific energy/ strain energy density, Editors Sih G.C.,Czoboly.E and Guillemot.F, pp 107~129. 5.5 Firrao.D. and Roberti.R. (1983). ‘Ductile fracture nucleation ahead of sharp cracks’. Metallurgical Science and Technology, No 1, pp. 5~13, June. 5.6 Akourri .O., Louah.M., Kifani.A. & Pluvinage.G.(2000). ‘The effect of notch radius on fracture toughness J1c’ Engineering Fracture Mechanics, pp 491~505. 5.7 Sumpter J. D. G. (1973). ‘Elastic-plastic fracture analysis and design using the finite element method’. PhD. Thesis University of London. 5.8 Morozov. E., Pluvinage.G. (1996).’Study of stress and strain concentration coefficient by path-integral’ Problems of Strength, Special Publication 96, pp. 53~64.

CHAPTER 6 STRAIN CRITERIA FOR FRACTURE EMANATING FROM NOTCHES _____________________________________________________________________ 6.1 Introduction Fracture emanating from notches can be modelled by the strain fracture criteria. Similarly as for stress or energy criteria, local and global approaches can be used. Based on the literature, it can be seen that both the strain and energy criteria are used less than those of the of the local and global stress approaches. This can be accounted for by traditional design codes use a design stress rather than a strain However this kind of criteria can be used particularly for situations where plastic deformation is strongly involved. In this case fracture occurs in a ductile manner or by plastic collapse. The transition between these two modes of failures depends of the size of the ligament ahead of the crack. When ductile fracture occurs, notch geometry becomes more influential on the fracture process. This is because the notch radius has a strong influence on the stress triaxiality and consequently on the local or gross strain. The description of the strain distribution at the notch tip can be made by introducing the notion of pseudo strain singularity. In the following chapter a local strain criterion for failure emanating from notches is described. In this criterion, the fracture process volume is governed by an average strain called the effective strain. This chapter also contains the influence of triaxiality and ligament size on critical local or gross strain. 6.2 Critical strain criterion for fracture emanating from notch 6.2.1 EVOLUTION OF THE CRITICAL STRAIN WITH NOTCH RADIUS The critical gross strain is an indirect measure of the average net stress in the ligament ahead of the notch with an additional contribution of the elastic strain in the un-notched section of the specimen. Due to the change of stress distribution at the notch tip with notch radius, the average net strain and consequently the gross strain is sensitive to U. To determine the influence of notch radius on the notch tip stress distribution, tests have been performed on axisymmetrical notched specimen made in XC 18 steel (French standard). The chemical composition of this steel is listed in table 6.1. % XC18

C 0.20

Mn 0.57

Si 0.26

S 0.019

P 0.029

Cr 0.08

Table 6.1 Chemical composition of the studied steel. 113

Ni 0.07

Mo 0.02

114

G PLUVINAGE

Material E (MPa) Re (MPa) Rm (MPa) k (MPa) n XC18 steel 196000 260 520 891 0,28 Table 6.2 Mechanical properties of the studied steel.

Mechanical properties such as Young’s modulus E, Yield stress Re, ultimate strength Rm and parameters k and n of the Ludwik’s law are given in table 6.2. The geometry of the specimens used is shown in figure 6.1 These specimens exhibit a notch of radius U in the range [0.25 ~1.2 mm].

U=

0.25 0.5 0.8 1.0 1.2

20

M10 25 90 Figure 6.1: Geometry of axisymmetric specimen with dimensions in millimetres.

The evolution of the critical strain Hf is plotted versus the notch radius in figure 6.2.Here it can be seen that the critical strain is a decreasing function of the notch radius which can be described by a polynomial function. 0.5

Critical strain Hf

0.4 0.3 XC 18 STEEL

0.2 0.1 0.0 0.0

0.2

0.4

0.6

0.8 1.2 1.4 1.0 Notch radius U(mm)

Figure 6.2: Evolution of the critical strain Hf with notch radius (XC18 steel).

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

115

The critical strain is determined from the change in diameter under the notch within the relationship: §D · (6.1) H f 2 Ln¨¨ 0 ¸¸ , ©Df ¹ D0 and Df are respectively the initial and final diameter. 6.2.2 EVOLUTION OF THE CRITICAL GROSS STRAIN WITH NOTCH DEPTH The geometry of the specimen also influences the critical gross strain; most notably the notch depth This has been studied on steel SENT specimens made in a ductile steel (French Standard E 36) having the following mechanical properties : yield stress Re = 323 MPa; ultimate strength Rm = 443 MPa. The relationship between stress and strain can be expressed by Ludwik’s law :

V

kHn ,

(6.2 )

where k is a material constant (k = 605 MPa) and n is the strain hardening exponent (n = 0,146). The geometrical characteristics of the specimens are listed in table 6.3. thickness width length ligament size notch radius

B = 5 mm W = 29 mm h = 90 mm b [14-24 mm] Uҏ= 0.5 mm

Table 6.3 : geometrical characteristics of the specimens

Two strain gauges are mounted on the specimen: the first one is remote from the ligament in order to record the critical gross strain Hcg and the second is located on the ligament itself in order to obtain a reference value of strain to be compared with the numerical results. The evolution of the critical gross strain versus the non dimensional notch depth can be seen on figure 6.3. Critical gross strain decreases quickly (zone I) and remains at a low and constant value for a/W greater than 0.25~0.27 according to the notch radius (zone II). This diagram indicates that for short notch, the specimen fails by general collapse. For a/W greater than 0.25 critical gross strain is near to the yield strain and plasticity develops along the ligament length. These experimental results show that the notch effect for ductile fracture appears for a limited range of notch lengths smaller than for a crack which fails in a brittle material. At small notch lengths (approx 0.25), a transition from general to local plastic collapse

116

G PLUVINAGE

occurs. This transition has been previously mentioned by Soete [6.1] who uses precisely this transition to define the critical defect size in a structure. Critical gross strain Hcg

ZONE I Plastic collapse

ZONE II Instability along ligament

0.1

0.0

Critical notch depth

0. 1

0.4 0.5 0.3 0.6 Non dimensional notch depth a/W

0.2

Figure N° 6.3: Evolution of the critical gross strain Hgc versus the non- dimensional notch depth (E 36 steel).

6.2.3 INFLUENCE OF STRESS TRIAXIALITY ON CRITICAL STRAIN FOR NOTCHED SPECIMENS Stress triaxiality plays an important role on the critical gross strain as reported MacClintoch [6.2], Zerek et al [6.3], Hancock et Mac Kenzie [6.4], Holland et al [6.5]. It is well known that the critical strain Hf exponentially decreases with the stress triaxiality according to the relation ship (6.3) : (6.3) H f H n  A. exp BE , A and B are constants (B = 1.5 according to [4] and 5.23 according [6.5]). Hn is the nucleation strain for voids which are the origin of the ductile fracture. El Magd [6.6] has proposed the following relationship: C H f Hn ,

E

(6.4)

where C is also a constant. For axisymmetrical specimens, triaxiality has been computed using the Bridgman’ formula 1 § D · E (6.5)  Ln¨1  0 ¸ . 4 ¹ 3 ©

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

117

Critical strain has been plotted versus triaxiality according to relationship (6.6) and is shown in figure 6.4. (6.6) H f  H n A. exp BE .

In the case of XC 18 steel and for static loading, A = 1.62 and B = 1.53. 0.3

Term (Hf-HN)

0.2

XC 18 STEEL

0.1

0.0

0.8

1.0

1.2

1.6

1.4

1.8 Triaxiality E

2.0

Figure 6.4: Evolution of the term (Hf-Hn) versus the triaxiality for XC 18 steel.

6.2.4 Influence of triaxiality and loading rate on critical strain The influence of loading rate and triaxiality on the critical strain can be considered for strain rate sensitive material which obeys the following stressístrain curve:

V eq



m n . H eq , K H eq H*

(6.7)

where Veq and Heq are equivalent stress and strain respectively, k, n and H are constants and m is the strain rate sensitivity exponent. In this case, the critical strain is given by a relationship due to El-Magd [6.6] (D is a constant).

Hf

D Hn 1 nm E

.

(6.8)

Critical strains obtained from results of static and dynamic fracture tests on notched specimens of XC 18 steel are plotted versus (E)(1/n+m) with n = 0.28 and m = 0.02, see

118

G PLUVINAGE

figure 6.5. Relationship (6.8) provides a good a fit of the experimental datas. Values of nucleation strain, constant D and correlation coefficient R are listed in table 6.4.

0.4

Term (Hf-HN) STEEL XC 18

0.3

STATIC DYNAMIC

0.2

0.1

0.0 0.0

0.3

0.6

0.9

1.2 Term E (1/n+m)

1.5

Figure 6.5: Evolution of the term (Hf-Hn) versus the parameter E)(1/n+m) for two loading rates for XC 18 steel.

Static Dynamic

Hn

D

R

0. 209 0. 214

0.286 0.184

0.987 0.975

Table 6.4: nucleation strain, constant D and correlation coefficient R for relationship (6.8).

6.3 Strain distribution at the notch tip

6.3.1 DESCRIPTION OF THE STRAIN DISTRIBUTION AT A NOTCH TIP IN A LINEAR DIAGRAM One example of the computed elasto-plastic strain distribution is presented in figure 6.6 Thickness B Width W height L ligament length b notch radius U

15 mm 80 mm 96 mm 40 mm 1 mm

Table 6.5: Geometrical characteristic of the Studied CT specimen.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

119

This strain distribution has been computed using finite element method. The distribution is computed for a C.T specimen having the following geometry: The material is a French standard steel E 550 having a yield strength of 572 MPa and ultimate strength of 684 MPa. The material obeys Ludwik’s law. The distribution decreases rapidly from a maximum strain Hmax given by: (6.9) H max k H . H g , where kH is the elasto-plastic strain concentration factor and Hg is the gross strain. This type of strain distribution has been modelled using different relationships which can be found in the literature For example, the relationship given by Griffith and Owen [6.7] is: § x· H pl H pl , max . exp¨¨  1.7 ¸¸ , U © ¹

(6.10)

where Hpl,max is the maximum strain at the notch tip. Xu [6.8] has extend this formula by replacing the coefficient 1.7 by a parameter c : § x· (6.11) H pl H pl , max . exp¨¨  c. ¸¸ , U¹ © where c depends upon the strain hardening exponent and the load level, c decreases with the load level. 0.07

Strain normal to notch plane (Hyy)

0.06 0.05 CT SPECIMEN STEEL E 550 Notch radius U = 1mm

0.03 0.02 0.01 0.0 0

4

8

12 16 Distance from notch tip (mm)

Figure 6.6: Example of the elastoplastic strain distribution at the notch tip of a CT sample (E 550 steel) ; notch radius 1 mm.

120

G PLUVINAGE

Usami (6.9) proposed a relationship for the elastoplastic strain distribution in a polynomial form. x 2 3 x  4º H max ª« 1 ».  . 1 1 . 1 (6.12) H yy U U 2 3 « 2 » ¬ ¼





6.3.2 STRAIN DISTRIBUTION AT THE NOTCH TIP IN A BILOGARITHMIC GRAPH A typical notch tip strain distribution is similar to that of the stress distribution when described in a bilogarithmic graph. The strain distribution can be divided into 4 zones: í Zone I, where the strain is at its maximum close to the notch tip and is related to the gross strain Hg and the elastoplastic strain concentration factor kH: (6.13) H max k H . H g , í Zone II is intermediate between zone I and Zone III, íZone III which can be described as a zone of pseudo-singularity for strain. In this area the strain-distance relationship has the following form : k U,H , (6.14) H yy D" 2 S r  where D” is a parameter characterising the power dependence of the relationship and kUѽH is the so-called notch strain intensity factor. ízone IV where the strain decreases rapidly The limit between zone II and zone III has for abscissa Xef,H the effective distance (for the strain distribution). It corresponds to the effective strain Hef. On the characteristic D" line of the pseudo singularity, the following product H yy . 2Sr is a constant and equal to the notch strain intensity factor. At the critical load:

H yy . 2Sr

D"



H ef . 2S X ef

D " k U , H ,

(6.15)

where ҏHef is the effective strain respectively and Xef,H the corresponding distance.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

Figure 6.7: Example of the elastoplastic strain distribution at the notch tip presented in a bilogarithmic graph.

Strain normal to notch plane (Hyy) 10 1 10-1 CT SPECIMEN

U = 1mm

10-3

U = 0.75 mm U =0.50mm U = 0.10 mm

10-4 10-5

10-4

10-3

10-2 10-1 1 Non dimensional distance r/b

Figure 6.8 Strain distributions at the notch tip of a CT specimen for different notch radii.

121

122

G PLUVINAGE

6.3.3 EVOLUTION OF THE PARAMETERS OF THE STRAIN DISTRIBUTION WITH NOTCH RADIUS The characteristic parameters of the strain distribution at notch tip are: the maximum strain Hmax, the effective strain, Hef , the effective distance, the exponent of the pseudo singularity D’’. . Critical effective strain Hcef (%) 12 CT SPECIMEN STEEL E 550

U= 1mm U = 0.5 mm U = 0 mm (crack)

8

4

0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Non dimensional notch depth a/W Figure 6.9: Evolution of critical effective strain as a function of the notch radius and non-dimensional notch depth.

The notch stress intensity factor is related to the last three parameters through relation ship (6.15). Figure 6.8 presents different strain distributions at the notch tip of a CT specimen with different notch radius. We can notice that strain distributions present a maximum strain which decreases exponentially with notch radius. The critical effective strain also varies with the notch radius and with the nondimensional notch depth as it can be seen in figure 6.9. It increases withҏU but also increases with a/W. A sharp transition in critical effective strain can be seen at an a/W = 0.4 for the notch radii 0.5 and 1 mm. The parameter D” characterising the pseudosingularity is not sensitive to the notch radius as we can see in figure 6.10. Furthermore, it decreases non linearly by 26% from a/W [0.17-0.52]. 6.3.4 LOCAL STRAIN FRACTURE CRITERION A local strain fracture criterion is also based on the concept of the fracture volume process which has been previously described in the case of the local stress fracture criterion. The limit of this fracture process is also the beginning of the strain pseudo singularity. This point

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

123

can be seen on Figure 6.6 and corresponds the point Xef,H. For the critical event, the strain for this abscissa is the critical effective strain.

Figure 6.10: Evolution of the parameter D” with notch radius and non-dimensional notch depth.

D" The product H c . 2S X ef , H is precisely the critical notch strain intensity factor ef , H which can be taken as a measure of the fracture toughness, i.e., D" . (6.16) K cU , H H cef , H . 2S X ef , H









In other words fracture occurs when the notch strain intensity factor reaches a critical value: (6.17) K U , H K cU , H . It should be also noted that the product of the strain and the distance to the power D’’ is constant along the line of pseudo singularity. In this respect, any error in the distance cannot affect the value of the critical notch strain intensity factor. 6.3.5 THE NOTCH DUCTILITY FACTOR This type of local strain criterion was proposed by Randall and Merkle [6.10] in 1973 where they define the fracture toughness of a notched specimen as the product of the local fracture stress and the square root of the notch radius. This analysis is based on Neuber's relationship: (6.18) k t2 k t . k t ,

124

G PLUVINAGE

where kt is the elastic stress concentration factor and kV and kH the elasto-plastic stress and strain concentration factors respectively. For an elliptical notch in a finite plate subject to tensile loading, the elastic stress concentration factor is given by the following formula: · § ¨1  2 a ¸. F V a , W ¨ U ¸¹ ©



kt

(6.19)

where a is the notch depth and U the notch radius. FV(a/W) is a geometrical correction factor. By applying Neuber's relationship, we find: § V ·§ H · ¨ I ¸.¨ I ¸ , ¨V g ¸¨H g ¸ © ¹© ¹

kt

(6.20)

where Hg and Hl are respectively the global and the local strains and Vg and Vl the global and local stresses. It is assumed that remote from the notch, the behaviour remains elastic:

Vg

, (6.21) E where E is the Young's modulus. Locally the following relationship can be found:

Hg

Hl

Vl El

,

(6.22)

where El is the ‘local strain modulus’, where it is assumed that the fracture criterion is given by : (6.23) H l H lc , H g H cg . The quantity Hl¥U is considered as a fracture toughness parameter. E c .H g . H l U 2 F V a W . a. El



(6.24)

The global critical strain Hcg is given by the linear fracture mechanics relationship: K *c Re . H y , (6.25) H cg F V a W . Sa





where Hy is the strain at yield and K *c the apparent fracture toughness. (6.26) H lc U C. K *c Re ,





FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

125

where C is a parameter which is a function of temperature and thickness. In this case, the Notch Ductility Factor (NDF) differs from the critical strain intensity factor by a constant D" D" NDF H c . X ef , H (6.27) K cU , H  2S ef , H





and also differs from the Randall and Merkle definition

 NDF R  M H cmax . U These definitions of the Notch Ductility Factor lead to the following differences: present definition Randall - Merkle Xcef ,H U

effective distance effective stress

Hceff

Hcmax

pseudo-singularity parameter

D"

0,5

Hcef (Xcef,H D"

NDF

Hcmax¥U

Table 6.6 Characteristics of the definitions of the Notch Ductility Factor.

D" The influence of the notch depth on the term H c . X ef , H can be seen in figure ef , H 6.11 and is compared to the Randall and Merkle Notch Ductility factor Hcl¥U.





Notch ductility factor

(mm)D ‘’

(mm)0.5 0.4

Hcef,H.(Xcef,H)D’’

0.07

Hcmax.U 0.5

0.06

0.2

0.05

0.1

0.04 0.03 4

6

8

10

12

14

0.0 16

Notch depth (mm)



Figure 6.11 : Influence of notch radius on notch ductility factor H c . U and term H c . X ef , H ef , H l

D"



.

126

G PLUVINAGE

The Notch Ductility Factor is generally 3.5 times greater than NDF R-M of Randall and Merkle and exhibits a similar or smaller scattering. It should be noted that units are different i.e. respectively mD” and ¥m. The variation of the NDF is plotted versus the non-dimensional notch depth for three notch radii (Figure 6.11). This parameter increases with notch radius and depends on the non-dimensional notch depth. 0.08

Notch ductility factor Hcef(Xcef)D’’ ; (mm)D’’ Notch radius U= 1 mm U= 0.5 mm U= 0 mm (crack)

0.07 0.06 0.05 0.04 0.03

0.27

0.47

0.02 0.1

0.2

0.3

0.4 0.5 0.6 Non dimensional notch depth a/W

Figure 6.12: Evolution of the Notch Ductility Factor with notch radius and non dimensional notch depth.

As we can see from figure 6.11, there are ranges of a/W values where the notch effect in ductile fracture is relatively small (0.27 ” a/W ” 0.47). It decreases when the ductility of the material increases. By examination of the value of the critical gross strain, we can see that for low a/W values this strain is higher than the yield strain and the structure is totally plastic. Failure occurs by general plastic collapse. For higher value of a/W (a/W > 0.27 in this case) plastic deformation is localised in the ligament and the notch effect appears. For values higher than 0.47, the plasticity is made up of two plastic zones: a tensile zone and a compressive zone. When the ligament is totally plastic the notch effect disappears. 6.3.6 INFLUENCE OF STRESS TRIAXIALITY ON THE LOCAL FRACTURE STRAIN The ductile fracture process can be divided in three steps: i) void nucleation from non-metallic inclusions; ii) void growth; iii) final instability of the ligament between the voids.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

127

The stress instability plays an important role on void growth and consequently on the local critical strain. Rice et Tracey [6.11] have shown that the growth rate of a spherical cavity of diameter R in a perfectly plastic material, is given by: dR 0.28 exp>1.5E @ . (6.29) RdH 1 STEEL XC18 static

0.8

dynamic 0.6 0.4 0.2 0

0

0.4

0.8

1.2

1.6

Stress triaxiality E Figure 6.13: Variation of the logarithm of the RC/RO ratio versus the triaxiality (notched axisymmetric specimen, XC 18 steel).

Barnby et al [6.12] proposed the following relationship: dR 0.111  1.893E d H p , R

(6.30)

where dHp is the plastic strain increment. The local critical strain is obtained from the diameter of the surface cups as measured from the fracture surface of an axisymmetric specimen. assuming that they have a spherical shape and the material incompressible. The average value of the statistical distribution of the size of the cups in a small area is called Rc ; the average value of the inclusion size is denoted R0. Based on these parameters, the local critical strain is given as: § Rc · ¸. (6.31) H lc Ln¨¨ ¸ © R0 ¹ The local fracture strain from the size of the ductile cups has been measured on the fracture surface of axi-symmetric specimens of XC 18 steel for dynamic and static loading. These values are plotted against the stress triaxiality termE as shown in figure 6.13. The lines plotted on this figure are based on the following relationship: (6.32) H lc F . exp HE .

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G PLUVINAGE

where F and H are constants which are function of both material and loading rate. The ratio of the local fracture strain and critical gross strain gives the value of the elastoplatic stress concentration kH . This value has been plotted versus the stress triaxiality as shown in figure 6.14.

3.5

Elastoplastic strain concentration factor ke dynamic

STEEL XC 18

static

2.5

1.5 1.00

1.25

1.50

1.75

2

Stress triaxiality E Figure 6.13: Variation of the elastoplastic stress concentration factor kH versus the stress triaxiality for static and dynamic loading (notched axi-symmetric specimen, XC 18 steel).

The elastoplastic strain concentration factor increases with stress triaxiality and has a value between 2 and 3. A mutual influence of the stress triaxiality and the critical strain has been seen by Shockey et al [6.13] and Barnby et al [6.12] and proposed the following relationship: Rc (Shockey et al). 1.07 E . H f , (6.33) R0 Rc R0

(Barnby et al).

The variation of the ratio Rc

R0 .H f

0.111  1.893E .H f

.

(6.34)

is plotted versus the stress triaxiality for the same

experimental data in figure 6.15. The regression lines in this figure have been plotted according to the following linear relationship:

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

129

Rc >L  M .E @.H f , (6.35) R0 where L and M are constants which depend on the material and loading rate.

Rc with the stress triaxiality for static and dynamic loading R0 .H f (notched axisymmetric specimen, XC 18 steel).

Figure 6.15: Variation of the ratio

6.4 Notch plastic zone The plastic zone includes the fracture process zone and produces a “screening effect“ to limited the energy flow coming from the entire part of the structure and providing the necessary fracture energy. This is one of the basic reasons for knowing the size of plastic zone at the notch root. More information can be found in the literature for the size of crack tip plastic zone rather than the notch root plastic zone. The plastic zone diameter Ry is given (for cracks) by the general formulae: Ry

E '.

K 2 , Re



(6.36)

where Re is the yield stress and K the stress intensity factor.ҏE' is a coefficient which varies in the range 0.30 ~0.39. E takes the particular value of 0.318 for Irwin’s formula [6.14], and 0.342 for Dugdale’s formula [6.15]. Irwin’s formula for the crack plastic zone is given by :

130

G PLUVINAGE

Ry

1

S

.

K 2 , Re



(6.37)

while Dugdale’s formula gives :

S 2 V 2g a

SK

. (6.38) 8 Re 2 8 Re 2 CT Specimens were made from XC 38 French standard steel having the following dimensions: width W = 80 mm, thickness B = 40 mm, notch length a = 40 mm. The notch angle was constant, < = 40°. The notch radius was varied having the following values U = 0.3; 0.5; 1 ;1.5 and 2 mm. The mechanical properties of the steel are listed in Table 6.7. Rp

E (GPa) Re (MPa) Rm (MPa) A (%) Young’s modulus Yield stress ultimate strength elongation 206 350 522 30

Z (%) Reduction in area 64

Table 6.7: The mechanical properties of the XC 38 French standard steel.

Each specimen was subjected to a load P =19000 N. This load corresponds to a net stress 317 MPa, according to relationship (6.39). This value is about 90 % of the yield stress Re according to table 6.7. ª 3W  a º P .«1  . (6.39) VN W  a .B ¬ W  a »¼ The plastic zone size was determined by two methods: a) metallographic investigation and b) computing by a Finite Elements method. These two methods are compared.

Figure 6.16: Luders ‘s bands at the notch root CT specimen, U = 0.3 mm, VN = 317 MPa. (G x40)

Figure 6.17: Luders’s bands in plastic zone. CT specimen, U= 0.3 mm, VN = 317 MPa. (G x10)

The plastic zone R zp y in the y direction is measured on the Luders’s band pattern (see figures 6.16 and 6.17). From Figure 6.18, it can be seen that the plastic zone size increases with notch radius Furthermore, its shape depends upon nature of the loading as shown in figure 6.19(a) and (B). For the elastoplastic case, the plastic zone shape lies more along the x axis and

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

131

is similar in shape to that revealed by metallographic examination. Good agreement can be found between experimental and computed evaluation of the plastic zone size. Plastic zone size (mm)

2.6 2.4 2.2

STEEL XC 38 2.0 1.8 1

0

2 3 Notch radius U(mm)

Figure 6.18 : Variation of the plastic zone size with notch radius.

D

C

1 mm

1 mm A) Plastic zone shape (Elastic behaviour)

B) Plastic zone shape (Elastoplastic behaviour)

Figure 6.19: Shape of the computed plastic zone for 4 level of stress and two mechanical behaviour. (Level of net stress 1: 350 MPa 2: 400 MPa 3: 450 MPa 4: 500 MPa) ; CT 40 specimen ; VN = 317 MPa ; U = 0.3 mm ; \ =40°.

In the elastic case, the typical ‘bean’ shape is similar to that associated with a crack. In this case the plastic zone is given according to Tresca’s plasticity criterion by: (plane strain)

R y T

K 2 . cos T , 2S Re 2

(6.40)

132

G PLUVINAGE

K2

R y T

(plane stress)

> 

@

T 2

T

. cos . 1  sin 2 2 2S Re 2

.

(6.41)

zp

The dimension of the plastic zone in the y direction R y increases with the notch radius as you can see in figure 6.18. Plastic zone size along x direction (mm) 4

U= 0.1 mm U= 0.2 mm

3

U= 0.3 mm

2

1

STEEL XC 38 0 0

100

200

300

400

Figure 6.20 : Size of the plastic zone versus the net stress VN. (CT Specimen ; notch radius U = 0,1 ; 0,3 and 0,5 mm ; notch angle < = 40°).

For the Von Mises criterion we get: 3 K2 . sin 2 T  1 Q 2 . 1  cos T (plane strain) R y T 2 4S Re 2

>

(plane stress)

R y T

K2 4S Re 2



>



3 . 1  sin 2T  cos T 2

@

@

(6.42)

(6.43)

Here the size of the plastic zone increases with the net stress in the x direction. This evolution seems insensitive to notch. The plastic zone size in the y direction can be expressed through the notch stress intensity factor by the relationship: K 2 . (6.44) Ry A Re



where the parameter a is less than 2. The variation of the parameter A with the elastoplastic stress concentration factor is presented in figure 6.2.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

133

1.2 1.0 0.8 0.6 0.4 Irwin’s solution for crack

0.2 0.0 0

5

10 15 Elastic stress concentration factor kt

20

Figure 6.21 : Variation of the parameter A with the elastoplastic stress concentration factor k t.

6.5 Conclusion

The critical gross strain is sensitive to the ligament size and the stress triaxiality. Plastic failure occurs for a ligament size less than a critical value which can lead to a new definition of the critical defect size. The strain distribution at the notch tip can be presented by a maximum and a notch strain intensity factor. The critical notch strain intensity factor can be used as a fracture toughness value. The notch ductility factor can be considered as another measure of fracture toughness.

134

G PLUVINAGE

REFERENCES 6.1 Soete. W. (1968). ‘An Experimental approach to fracture initiation in structural steels’. 6.2 Mac Clintock F.A. ,Journal of Applied Mechanics 35, p 363. 6.3 Zurek A. K and al. (1988). Congrès Dymat, Journal de Physique C3 49 ", p 269, 6.4 Hancock. J.W. and Mac Kenzie. A.C. (1976). Journal of Mech. Physic. of Solids 24 , pp 147-169, 6.5 Holland. D., Hallen. A. and Dahl.A. ( 1990).Steel Research, pp. 504~506. 6..6 El Magd. E L. (1995).’ Dynamic Ductibility of Metallic Materials’ Dymat, Munich. 6.7 Griffith J.R. and Owen.D.R.J. (1972). Journal of Mechanics and Physics of Solid,19,p 419. 6.8 Xu Kewein, He Jiawen. (1992).‘Prediction on notched fatigue limits for crack initiation and propagation.” c l. s, Vo ° 41, N 3, pp 504~410, 6.7 Griffith.J.R and Engineering Fracture Mechani 6.9 Usami. S. (1985). ‘Short crack fatigue properties and component life estimation’. Current Research on Fatigue Cracks, Edited by Tanaka, T.;The Soceity of Materials Sciences, Kyoto. 6.10 Randall. P. N and Merkle. J.G. (1972). ‘Effects of Crack Size on Gross-Strain Crack Tolerance of A533-B Steel’. Journal of Engineering for Industry, p 935~941, August. 6.11 Rice. J .R and Tracey .D .M. (1969). Journal of Mechanics and Physics of Solids ,Vol 26,pp 163~186. 6.12 Barnby J .T. and al.(1984). ‘On the void growth in C-Mn structural steels during plastic deformation’. Int Journal of Fracture 25, pp 271~284. 6.13 Shockey D .A and al. (1980). ‘Computational modelling of microstructural fracture process in A 533 B pressure steel NP 1398’. Final Report EPRI. 6.14 Irwin. G. R. (1948).‘ Analysis of stresses and strains near the end of a crack traversing a plate” Trans Asme Journal of Applied Mechanics,24, pp 361~364. 6.15 Dugdale .S. (1960). ‘ Yielding of steel sheets containing slits’.Journal of Mechanical Physics of Solids,vol.8,pp 100~104.

CHAPTER 7 THE USE OF NOTCH SPECIMENS TO EVALUATE THE DUCTILE TO BRITTLE TRANSITION TEMPERATURE; THE CHARPY IMPACT TEST ______________________________________________________________________ 7.1 History of the Charpy impact test Charpy’s work relating to impact testing was presented for the first time in 1901 at the Budapest Congress of the International Association for Testing of Materials [7.1]. The paper of Charpy, a principal engineer at the Saint Jacques factory of Montluçon (France), was written in French and entitled ’ Note sur l’essai des métaux à la flexion par choc de barreaux entaillés’. Previous works in the field of mechanical testing have foundthe lack of correlation between the static and dynamic test results. This conclusion appears first mentioned by Mr Lebasteur in his book entitled ‘Les métaux à l’Exposition de 1878’ in which he concluded: ‘It is impossible to get full results of the strength of materials without any impact test ‘. Earlier at the French Commission of Testing Methods in 1892, Mr Le Chatelier proposed the use of notched specimens for themeasurement of the resistance to fracture. This type of specimen has been used from 1893 by Mr Auscher, a ship building engineer at Indret (France). The geometry of the specimens was a 20 mm square section with a one millimetre deep triangular notch on each of the four faces. Specimens were clamped, and drop weight tests done with a 1 kilogram hammer falling successively from different heights at the free end of the specimen. This end was located 100 mm from the clamped section. This method has the disadvantage that it requires several tests in order to obtain a consistent result. The idea of measuring the residual force after fracture and consequently the work done for fracture, was introduced by Russel in the USA and by Fremont in France. Russell presented a paper in 1897 at the American Society of Civil Engineers in which he described the use of a pendulum. The work done for fracture was established from the difference between the initial and the final height. Fremont in the ’Bulletin de la Societé des Ingénieurs civils’ in November 1897 presented a device in which the residual energy after impact by the pendulum was measured from the compression of a spring. The pendulum built by Charpy is presented in figure 7.1. The weight of the pendulum was 50 kilograms and the distance between the axes to the knife was 4 metres. The pendulum was seated on a block of masonry of five cubic metres. The specimen was seated on two supports made in a plate of 1,600 Kilograms. After impact the height was measure directly because the velocity of the pendulum was slow and the difference of height between arrival and departure multiplied by the weight of the pendulum gives a measure of the work done for fracture. Dissipation energy was evaluated by an experiment without impact (without using any specimen) and found to be less than 2% of the total energy. An experimental correction was proposed in order to have a negligible error after correction 135

136

G PLUVINAGE

The error owed to kinetic energy was considered to be less than 1% of total energy the weight of the specimen was less than 1/150 of the weight of the pendulum. Fragments were thrown to a distance between only 2 or 3 metres. G Charpy has estimated that he can measure the work done for fracture with an error less than 1%.

Figure 7.1: A device for impact testing proposed by Charpy having impact energy of 250 kgm [1].

He also proposed to call the work done for fracture for an infinite small thickness ‘Resilience’ and he mentioned that this word was also used by Russel. The unit is expressed as kilogram-metres per square centimetre. It seems that at that time the concept of brittle fracture was not very clear. Charpy pointed out that the test on notched specimens is not a test for brittle fracture but rather a test which allowed the classification of metals either as having a high resilience or a low resilience. The term ‘brittle metal’ was reserved for metal having different behaviour under dynamic or static loading. However, despite the introduction during the 1960s of mechanical fracture testing to measure the resistance to crack growth, the Charpy impact test remains, as it gives a simple inexpensive method of classifying a materials resistance to brittle fracture. Nowdays, the tendency is also to use this test as a measure of fracture toughness. Comparison of the two methods requires the consideration of two major differences: i) Charpy tests use a notched specimen, fracture mechanics tests use a pre-cracked specimen (but some Charpy specimens are now pre-cracked), ii) fracture mechanics testing is generally performed using quasi-static loading conditions, Charpy tests are dynamic tests.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

137

7.2 Stress distribution at notch tip of a Charpy specimen 7.2.1 CHARPY SPECIMEN GEOMETRY Since the above studiestok place different kinds of Charpy specimens have been used. The most well known are presented in figure 7.2 and in table 7.1. In this table the specimen notch is defined by the notch radius U the notch angle \ and the notch depth a. A crack is considered as a special case of a notch with notch radiusU = 0 and notch angle \ = 0. 2 mm

10 mm

10 mm 55 mm CHARPY V

2 mm

5mm CHARPY U

MESNAGER Hard material pin

5mm KEY HOLE

SCHNADT

5 mm PRECRACKED Figure 7.2: Major types of Charpy specimens.

Specimen type Charpy V Charpy U Mesnager U Key hole Schnadt Precracked

notch radius U 0.25mm 1 mm 1 mm 1 mm 0.025 mm 0

notch angle \ ° 0° 0° 0° 45 ° 0

notch depth (a) 2 mm 5 mm 5 mm 5 mm 2 mm 5 mm

Table 7.1 Geometry of major types of Charpy specimen notches.

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G PLUVINAGE

7.2.2 STRESS DISTRIBUTION AT THE NOTCH TIP OF DIFFERENT CHARPY SPECIMENS MADE IN MILD STEEL Stress distributions have been determined for the following geometries: Charpy U, Charpy V, Schnadt, and pre-cracked Charpy, assuming that the material strain hardens and obeys to the following stress strain law: V K. H n , (7.1) where K is the strain hardening coefficient and n the strain hardening exponent (K = 737 MPa; n = 0.12). The stress distribution (for the stress normal to notch plane Vyy ) at the notch tip for the three types of specimen (Charpy U,V and Schnadt) are presented in Figure 7.3. It can be seen that the maximum stress is higher for Schnadt specimens (highest notch acuity) and the lowest value is obtained from the U specimen (greatest notch radius). The position of this maximum stress Xm moves far away from the notch tip when the notch acuity decreases. In Table 7.2 the value of the elasto-plastic stress concentration factor kVis summarised for three loads, kVLV defined as the ratio of the maximum stress to the net stress. kV

700

V max . VN

(7.2)

Stress normal to notch plane Vyy (MPa)

600

P = 1500 N

Schnadt

500 Charpy V

400

Charpy U

300 200 100 0 0.0

0.2

0.3

0.4

0.5

0.6

Distance from notch tip (mm) Figure 7.3: Stress normal to notch plane Vyy versus distance to the notch tip for three kind of Charpy specimen geometries (Schnadt, V and U).

It should be noted that the elastoplastic stress concentration is higher when the notch radius is small and is practically independent of the load level (exception of Charpy U).

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

Load

(Charpy kV (Charpy kV V) U) 5.97 1.72 5.88 2.066 4.99 1.95

kV (Schnadt) 9.57 9.55 9.00

1500 N 2500 N 3500 N

139

Table 7.2: Elastoplastic stress concentration for three types of Charpy specimens.

Load 1500 N 2500 N 3500 N

Xm (mm) Schnadt Xm (mm) Charpy V Xm (mm) Charpy U 0 0 0.14 0.044 0.045 0,17 0.05 0.14 0.62

Table 7.3: Position of the maximum stress for three types of Charpy specimens.

Table 7.3 presents, for different loading conditions, the position of the maximum stress Xm. From this table we note that the position of maximum stress moves away from the notch tip when the notch radius increases. The elastic and elastic-plastic normal stress distribution at the notch root exhibits a decreasing dependence with distance from the notch tip that is relatively complicated. Figure 7.4 presents in a bi-logarithmic graph, the stress distribution at the notch tip of a Charpy V specimen. As can be seen in this figure, the stress distribution is of classical form as previously described in Chapter 4. We can divide this stress distribution into 4 regions. from which we find the pseudosingularity in region III. log Vyy STEEL Cr.Mo.V P = 13,1 kN

3.0 D 2.5

1

2.0

I

II

III

IV

1.5 1.0 -1.2

-0.9

-0.6

-0.3

0

0.3

0.6

Logarithm distance (log r) Figure 7.4: Stress distribution at notch tip of a Charpy V specimen (Material Cr,Mo,V steel, load : 13.1 kN).

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G PLUVINAGE

7.3 Local stress fracture criterion for Charpy v notch specimens The local fracture criterion described in Chapter 4.5 is applied in the case of a Charpy V specimen This local stress fracture criterion requires two parameters: the effective distance Xef and the effective stress Vef. A graphical procedure for the determination of Xef described previously is applied. The effective distance is related to the minimum value of the relative stress gradient F. Charpy V notch specimens made of a CrMoV steel a with fine carbide (FC) microstructure (yield stress 771 MPa) were tested statically in bending at a selected temperature in the lower shelf region. The tensile stress distribution at the notch was calculated using FEM. A 2D model, was used under plane strain conditions for the elastic plastic analysis. The flow behaviour was computed using incremental plasticity adopting a Von Mises criterion. Finite element calculations were made using the ‘CASTEM 2000” package. The correlation between experimental and calculated results was good. Several tests were carried out at room temperature indicating a fracture load in the range [10.2-14.4 KN] which leads to an effective distance in the range [0.300~0.436 mm]. We note that this distance is greater than the microstructural unit (grain size for example). The effective stress is defined as the average stress inside the fracture process zone. For this material the mean value of the effective stress is 1223 MPa which can be compared to the average maximum local stress at fracture Vmax of 1310 MPa. The fracture toughness is given by : K cU 3.5



V ef . 2S X ef

D

(7.3)

log Vyy STEEL Cr.Mo.V Pc = 13,1 kN

F Vyy

2.5

Minimum of relative stress graddient

log Xef 1.5 Xef = 0.380 mm -1.0

-0.8

-0.6

-0.4 -0.2

0 0.2 0.4 0.6 0.8 Logarithm distance (log r)

Figure 7.5: Stress distribution at the notch root together with the relative stress gradient versus distance from notch tip for a fine carbide Cr Mo V rotor steel.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

141

The exponent of the pseudo-singularity D can be obtained according the formula proposed by Williams [7.2]. §\ · \ 2 \ 3 (7.4)  0.853 D \ 0.5  0.089¨ ¸  0.442 S S ©S ¹





The exponent D depends on the notch angle <. For a Charpy V notch specimen < = 45° and D is equal to 0.492 i.e., close to 0.5. The fracture toughness measured is equal to 62.7 MPa .m 0.492.. We can notice that the units are different from MPa¥m. 7.4 Influence of notch geometry on brittle-ductile transition in Charpy tests The Charpy energy is an increasing function of temperature. An example of the evolution of the impact resistance using a Charpy U notch specimen of a mild steel is shown in figure 7.6. A lower shelf region occurs at low temperatures where the resilience is low and the fracture appearance is brittle and with no macroscopic plastic deformation at the notch root. At high temperatures, the impact resistance is high and the fracture surface is ductile with contraction occurring at the notch root. The increase from low to high temperature leads to a transition from brittle to ductile fracture. Here the fracture appearance is mixed with a central brittle zone. Experimental data can be fitted using the following mathematical function : ª 1 º .T  A3 » (7.5) K CV A0  A1. tanh « ¼ ¬ A2 where KCV is the Charpy V impact resistance, T temperature in °C and A0, A1, A2, A3 are empirical constants.

20

Charpy impact resistance or resilience KCU (DaJ/m2)

15 Upper shelf 10

transition STEEL XC 12

5 Lower shelf 0

-200

-150

-100

-50

0

50 Temperature (°C)

Figure 7.6: Evolution of the resilience Kcu of mild steel (XC 12) with temperature.

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G PLUVINAGE

The temperature at which this transition occurs is an important tool for design against brittle fracture and the use of the Charpy test is an inexpensive method to determine this point. The transition temperature can be defined in three ways: ífor a given level of resilience for example 27 DaJ/cm2; íat the mid level between the lower and upper shelf regions; íat 50 % of crystallinity. The transition temperature is not intrinsic to the material as we can see on figure 7.7. In this figure, we can compare the work done for the fracture-temperature curve using the same steel but with a V notch (KCV) and U (KCU) respectively. From figure 7.7 we can see that for a Charpy V specimen, the transition temperature for brittle to ductile fracture is shifted from higher temperatures and the level of the ductile plateau is decreasing. Charpy energy (Daj)

20

STEEL ST 52-3 15

10

U notch V notch

5

0

-80

-60

-40

-20

0

20

40

Temperature (°C) Figure 7.7: Charpy Energy versus temperature for a Charpy U and Charpy V specimens, (Steel ST52-3).

Tests have been carried out using a steel St 52-3 [7.3] having the following chemical composition: C 0.21

Mn 1,25

Si 0,43

P 0,021

S 0,032

Al 0,01

Cu 0,37

Ti 0,01

Table 7.4: chemical composition of steel St 52-3 (weight %).

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

143

The transition corresponds to an initiation transition, i.e., brittle fracture initiates at a short distance behind the notch root Xef , ductile initiation at a distance greater than the effective distance and corresponding to the point of maximum triaxiality XE,max (Figure 7.8). It is well known that ductile fracture is sensitive to stress triaxiality as shown by Rice and Tracey [7.5]. The stress triaxiality is defined as the ratio of the hydrostatic stress to the equivalent Von Mises stress

E

Vm . V eq

(7.6)

The critical fracture strain increases exponentially with E. For this reason, ductile fracture initiates at the point of maximum triaxiality which changes with temperature. When the temperature increases, this point moves toward the notch tip and the brittle part of the fracture surface decreases. This ductile initiation is following by brittle propagation. At high temperatures the second transition appears and corresponds to a propagation transition. When the conditions of constraint at the tip of the running fibrous crack are sufficient, brittle fracture initiates. When these conditions cease to be satisfied because of the increasing temperature, the fracture surface becomes totally fibrous. The role of triaxiality on fracture initiation in Charpy test has been discussed independently by several authors [7.3],[7.4].

1000

Relative stress gradient F and stress triaxiality E

Log Vyy

F Stress triaxiality

E 1.5

500 Stress distribution 0

STEEL XC 12 P = 3500 N

Relative stress gradient

Charpy V

100 0.01

Xef 0.1

XE,max 1

1

10-5 0.5 10

Logarithm distance (log r) Figure 7.8: Effective distance Xef (point of brittle initiation) and point of maximal triaxiality (point of ductile initiation) in a Charpy V specimen (XC12 Steel) [7.4].

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7. 5 Instrumented Charpy impact test 7.5.1 DESCRIPTION OF INSTRUMENTED CHARPY IMPACT TEST The instrumented Charpy pendulum was developed during the sixties [7.6]. The use of pre-cracked Charpy specimens has allowed the connection with Fracture Mechanics concepts. The classical impact pendulum measures the energy used for fracture. By the addition of instrumentation to this equipment it is possible to record the load and the displacement versus time. The force measurement is carried out by strain gauges bonded to both sides of the hammer. These strain gauges are connected to a voltage supply and amplifiers. The calibration of the electrical signal derived from the strain gauges is made in two ways: -i) static calibration by loading the instrumented tup by compression on a mechanical testing device; -ii) dynamic loading based on the area under the registered load displacement curve until critical displacement dc being equal to the work done for fracture; dc U (7.7) ³ Pt Gd . 0 The velocity of the hammer is obtained by measuring the time interval between two pulses. These pulses are produced when two pins on the hammer go through an optical trigger device. Load (kN) 10

Load (kN) 10 DUCTILE FAILURE

5

Fm Fgy Fiu

5

0 0

1

2

Fa

3 Time (ms)

0 0

Figure 7.9: Example of recorded Charpy impact test data for a fully ductile failure .

1

2 Time (ms)

Figure 7.10: Example of recorded Charpy impact test data for a failure in the ductile brittle transition.

The displacement is measured via an angle measurement device and converted into linear displacement knowing the velocity of the hammer. The time to fracture initiation is generally measured by a magnetic emission probe [7.7] located near the notch root of the specimen. Load, displacement and emission probe

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

145

signals are recorded in on a multi channel digital memory scope connected to a computer and a printer. Two examples of recorded data are presented in figure (7.9 and 7.10). After a linear increase and several load oscillations owed to reflecting shock waves, the load increases to a maximum. When fracture occurs in the ductile-brittle transition regime, initiation appears after the maximum load and is characterised by a sudden drop. The final part of the diagram corresponds to final ductile tearing. The following parameters are registered on figure (7.10): Fgy load at general yielding; Fm load at maximum; Fiu load at the start of brittle propagation; Fa.load at the end of brittle propagation. 7.5.2 FRACTURE TOUGHNESS MEASUREMENT ON PRECRACKED CHARPY SPECIMENS Fracture toughness as measured from Charpy specimen increases with temperature according to a classical fracture toughness transition curve characterised by lower and upper shelf regions separated by a transition regime (Figure 7.11). In the lower shelf region, crack initiation occurs without macroscopic plastic deformation. Time to initiation is relatively short and can be more than the time necessary for load stabilisation after several load oscillations due to shock wave reflections on the specimen surfaces.

Fracture toughness

CLEAVAGE

KId, KJC, KJi

Impact response curve

CLEAVAGE/ DUCTILE KId, KJC, KJi

Standards

DUCTILE

KJi

Dynamic J-R curve

Temperature (°C) Figure 7.11: Schematic presentation of a ductile to brittle fracture toughness transition curve.

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The period of these oscillations is equal to W. According to Kaltoff [7.8], it is assumed that the time necessary for equilibrium is greater than 3W. In this case, the quasi static approximation to measure the dynamic fracture toughness KId according to ASTM standard is used : Pc . F P a W . (7.8) K Idc B W If the ’3W’ criterion is not satisfied, the impact response curve method from Kalthoff is used and the dynamic fracture toughness can be determined using equation (7.9) T v 0 .t" ,

K Id

(7.9)

where t" = f (t’) is given in the reference tales (7.8) and R is given by 2º ª (7.10) t" t c .«1  0.62§¨ a 0  0.5 ·¸  4.8 a 0  0.5 » , W W © ¹ ¬ ¼



R

301.



126 § 0.276.C · ¸ ¨1  ¨ 8.1*  9 ¸ 10 ¹ ©

(7.11)

where C is the compliance of the hammer in m/N and R in GN/m 2.5, Pc is the critical load and FP a geometrical correction. In the transition region crack initiation usually occurs after significant plastic deformation. The fracture toughness measured is obtained through the total work for fracture Uc (KId or J Id) or through the work until initiation Ui (Kid or J id). The dynamic critical value of the energy parameter JId is divided into an elastic and a plastic components (7.12) J Id J el  J pl . The elastic component is derived from the dynamic critical stress intensity factor Kcd: K 2Id . 1  X 2 (7.13) , J el E





and the plastic component from the plastic part of the wok done for fracture Upl : 2 U pl , (7.14) J pl B.W  a 0 where a0 is the initial crack length. By approximation JId is given from the total work done for fracture:

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

J Id #

2U c , B.W  a 0

147

(7.15)

The relevant KId value can be determined for plain stress conditions using equation (7.15). K Jd

E J Id ,

(7.16)

Similarly the fracture toughness Jid is given by: 2U i , Ji B.W  a 0

(7.17)

and the fracture toughness Kid is given by : K id

E J id .

(7.18)

In the upper shelf region fully ductile behaviour is observed. Characterisation of the material resistance is made possible by determining the dynamic JíR curve and obtaining the values J0.2d or Jid. The onset of stable crack extension is then measured via additional technique such as strain gauges, stretch zone measurement, COD or magnetic emission. 7.5.3 INFLUENCE OF SPECIMEN NOTCH RADIUS ON INSTRUMENTED CHARPY IMPACT TEST RESULTS The steel studied is a cast steel used for nuclear waste containers. The chemical composition is given in table 7.4 and the mechanical properties in table 7.5. Weight %

C

Si

Mn

Cr

Mo

Ni

Cu

Cast Steel

0, 09

0,3 7

1, 18

0,1 2

0,0 3

0,2 9

0,2 9

Table 7.4: Chemical composition of the steel studied.

Cast steel

Yield stress (MPa) 375

Ultimate strength (MPa) 478

A% 31,7

KCV +20°C DaJ / cm2 8

Table 7.5: Mechanical properties of the studied steel.

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G PLUVINAGE

The studied specimens are Charpy U specimen with different notch radius given in table 7.6. 0.7 1 2 2.5 U(mm) 0.13 0. 25 0.4 Table 7.6: Values of the notch radius used for Charpy U specimens.

Force Fm and Fgy (kN) 5

4

Fm

CHARPY U Cast Steel

Fgy

3 2 0

1

2

3

Notch radius (mm) Figure 7.12: Example of variation of load Fm and Fgy with notch radius for Charpy U specimen (Cast steel at room temperature).

This material is generally fully ductile and the following parameters have been measured ,load at general yielding Fgy ,load at maximum Fm, total work done for fracture Uc, and energy dissipated Um until maximum force is reached.

50 40 30

Maximum energy for fracture (J) CHARPY U Cast steel

20 10 0

Critical radius 0

1

2

3

Notch radius (mm) Figure 7.13: Evolution of energy until maximum force versus notch radius.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

149

As can be seen from figure 7.12, few difference appears between the values of the load Fm, Fgy for different notch radius. The variation of energy up to maximum force is presented in Figure 7.13. The variation is linear and increases up to a notch radius of about 1.2 mm after which a constant value is obtained. Total energy for fracture (J) 90 60 CHARPY U Cast steel 30

0

1

2

3

Notch radius (mm) Figure 7.14: Evolution of total energy for fracture versus notch radius.

From figure 7.14, the energy for fracture energy values exhibit a large amount of scatter with a low linear-regression analysis correlation factor. However, it appears than below notch radii less than of 1 mm a constant value of energy is obtained. Fracture toughness has been obtained from relationship: Uc (7.19) J Id K Bb where B.b is equal to 5.10-5 m2 , K varies with the notch radius according to Figure 5.9. According to [7.11] for high notch radii, the results exhibit a linear relationship passing through the origin. For low notch radii values of JIc are considered constant, figure 7.15. The constant value J*I c is equal to 1.,6 MJ/m2. The notch sensitivity factor DJ has been found to be equal to Dj = 1.23 GJ/m3 at the critical notch radius to 1,3 mm. 7.6 Equivalence fracture toughness KIc and impact resistanceKCV The Charpy impact test is a quick and inexpensive way of identifying the brittleness of a material. The transition temperature determined from this test can be a tool for the design against brittle fracture. However, this test cannot give directly the acceptable or the critical defect size. For this it is necessary to use a fracture toughness approach and determining the fracture toughness KIc. It is however to gain advantage of the use of the inexpensive and rapid Charpy test) and obtain a fracture toughness via a correlation using the Charpy energy KV (in Joules).

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G PLUVINAGE

4

Dynamic fracture toughness (MJ/m2)

CAST STEEL 3

2 JId = 1.6 MJ/m2 DJ = 1.23 GJ/m3

1

Critical notch radius Uc = 1.3 mm

0 0

2 Notch radius U (mm)

1

3

Figure 7.15: Dynamic fracture toughness JIc for a cast steel versus notch radius. Values of the notch sensitivity index.

This kind of correlation was first proposed by Barsom and Rolfe [7.9]. The correlation was different for the lower shelf; 2 K Ic 0.0225 K V 1.5 , (7.20) Re



to that of the upper shelf :

 K Ic Re

2

ª §K 0.62 «100¨¨ V ¬« © Re

· º ¸¸  1» . ¹ ¼»

(7.21)

A similar kind of correlation was proposed by Sailors and Corten [7.10]:

K Ic 14.6 14.6 K V .

(7.22)

The transition temperature is also sensitive to the stress state (for example, the transition temperature is lower for a specimen with a short crack than for a long crack, and lower for a long crack specimen than for a V notch specimen). For this reason it is necessary, in addition, to convert Charpy energy and fracture toughness to shift the transition curve for a given value of temperature. This method has been introduced in correlation proposed by Sanz et Al [7.12] and Wallim [7.13] Charpy V impact resistance Kcv (in Joules/cm2) may be converted to KIC using the Sanz and al correlation K Ic 19 K CV .

(7.23)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

151

The shift of the transition temperature is considered for of the particular impact resistance of 28 Joule/ cm2 according to . ' T t  K Ic

0.4 ' T t  K 28 .

(7.24)

An example of such a shift of transition temperature is given in figure 7.16 for a construction steel with a transition temperature T.K28 = -30 °C. The Wallim correlation [7.13] further takes into account the shift of the transition temperature T100 defined at a conventional level of the fracture toughness (100 MPa —m) and the transition temperature defined for CHARPY energy T27J at the conventional level of 27 J for the fracture energy. (7.25) T 100 T 27 J  18qC . Resilience (J/m2)

Fracture toughness (MPa—m)

200

KCV KIc

150

100

Shift = -18°C

50

0

-60

-40

-20

0

20

40

60

Temperature (°C) Figure 7.16: Conversion of impact resistance Charpy V into Fracture toughness KIc according to Sanz et al [7.12].

The concept of design presented in the last version of Eurocode 3 part 2 (January 1957) [7.14] considers the brittle assessment in the transition temperature and consists of a comparison of the fracture toughness Kmat of the material with the applied stress intensity factor KI,eq.

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G PLUVINAGE

K mat

K I , eq

k R6  U

,

(7.26)

kR6 is the value of the non-dimensional stress intensity factor according to the CEGBR6 method option 2 and U a plasticity correction factor according to the same method. K I,eq is the equivalent stress intensity factor of a structural component taking into account the design stress Vd and the design value of the defect ad K I , eq

M k .V d . S a d . F V a W ,

(7.27)

where FV(a/w) is the geometrical correction factor and MK is the correction factor to take accounting for the stress concentration factor. Kmat is given by: K mat

ª § ·º § T  T 100 · ¸¸  10 ¸¸» . (7.28) 20  «70.¨¨ exp¨¨ © 52 ¹ «¬ © ¹»¼

Fracture toughness KIC has been measured [7.15] on pipe steel having the following chemical composition: Ni Cu 0.29 0.29

C 0.09

Mn 1.18

Si 0.37

P 0.01

S 0.025

Cr 0.12

Mo 0.03

Table 7.7 Chemical composition of pipe steel.

Mechanical properties of this steel at 20 °C are listed in table 7.8. Yield stress Re 460 MPa

Ultimate strength Elongatio Rm n A% 630 MPa 24

Impact resistance (T-L) 17,5 J/cm 2

Table 7.8 Mechanical properties of a pipe steel.

The T100 transition temperature has been estimated at T100 = -51°C. The curve KIC = f (T-T100) has been plotted on the same graph along with the curve Kmat = f(T-T100) (see figure7.17). It can be seen that below the transition temperature, the results obtained from the experiments are close to the standard Kmat =f’(T-T100) curve given by Eurocode 3. At temperatures above the transition, the Eurocode Kmat curve overestimates the fracture toughness of the material. Charpy V energy has been measured at different temperatures and the transition temperature T27J has been

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

153

estimated to T27J = - 129°C. Charpy V energy has been converted into KIC using the Sanz correlation [7.12]. The values obtained have been plotted versus the temperature (T-T100) on the same diagram figure 7.17. In this case, the correlation is relatively poor. Fracture toughness (MPa—m)

PIPE STEEL 200 Eurocode Reference curve 100 KIc KCV 0 -200

-100

100 0 Relative temperature (T-T100) (°C)

200

Figure 7.17: Variation of fracture toughness with temperature (T-T100), Comparison with Eurocode reference curve [7.14]. Comparison with fracture toughness derived from Charpy energy.

The correlation between Charpy energy and fracture toughness suffers from the following problems: i) the dependence of a stress singularity is a 1/—r type and the dependence of a pseudostress singularity on a V notch with an angle of 45° is 1/r 0,49 (D has been computed according to Williams’ solution); ii) the stress state (triaxiality or Q factor) is different for a crack and a notch and the effective stress is sensitive to the stress-field, iii) fracture toughness is measured under static conditions while Charpy energy is measured under dynamic condition, i.e. the stress-strain curves are different. 7.7 Conclusion

Since the introduction of the famous Charpy impact test (1901) its procedure has been subject to several modifications. During the 1960’s, the nature of the notch acuity was strongly debated. and a sharper acuity was recommended. Atthe same time, the introduction of linear elastic fracture mechanics led to the use of pre-cracked Charpy specimens. However the introduction of this kind of specimen has never supplanted the use of the traditional Charpy V notch. The initial U notch recommended by Charpy has been progressively discarded.

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In the 1970s, the adoption of the fracture mechanics concept led to the evaluation of fracture toughness via the use of Charpy impact tests in terms of the critical stress intensity factor or the critical strain energy release rate. This approach requires that the critical load is suitably evaluated and the free oscillations experienced by the specimen during impact are understood. The recent introduction of Notch Fracture Mechanics has provided a promising tool for this problem. In this approach, there is no fundamental difference between fracture emanating from a notches or a crack. However, the effective fracture stress is sensitive to stress field conditions and particularly to relative stress gradient. It is interesting to notice that Charpy implicitly suggested that the specific work for fracture is a measure of fracture toughness. He termed this Resilience. This parameter is now realised as an energy fracture criterion in the form of JIC.

REFERENCES 7.1 Charpy.G . (1901). ‘Note sur l’essai des métaux à la flexion par choc de barreaux entaillés’. Association Internationale pour l’essai des matériaux Congrès de Budapest. 7.2 Williams. M. L. (1952). ‘Stress singularity resulting from various boundary conditions in angular corners for plates in extension’. Journal of Applied Mechanics. Vol. 19, N° 4, pp 526~528. 7.3 Toth.L.(1978).‘Rissbildungs und Ausbreitungsarbeit hinsichtlich der kerb-geometrie von kerbschlagbiegeversuchen’. Publications of the Technical University for Heavy Industry Series C, Machinery, Vol 34, pp31~47. 7.4 Pluvinage. G and Montariol .F. (1968) ‘Contribution à l’étude des transitions de résilience dans le cas d’un acier doux’. Mémoires scientifiques de la Revue de Metallurgie , LXV, N°4, , pp 297~308. 7.5 Rice. J. R and Tracey. D .M. (1969). Journal of Mechanics and Physics of Solids ,Vol 26, pp 163~186. 7.6 Ireland .D .R. (1976).’Critical Review of Instrumented Impact testing”, Proceedings International Conference on Dynamic Fracture Toughness, London, pp. 47~57. 7.7 Lenkey .G. (1999).’On the determination of Dynamic Fracture Toughness Properties by Instrumented Impact. Testing’. Prodeedings ASTM STP 1380. 7.8 Kalthoff. J.F.(1995).’Concept of Impact Response Curves’. ASM handbook, Volume 8, ASM, pp 269~271. 7.9 Barsom. J. M. and Rolfe.S .T . (1977). ‘Fracture and Fatigue Control in Structures’ Prentice-Hall. 7.10 Sailors.R .H. and Corten. H.T. (1972). ‘Relation between material fracture toughness using fracture mechanics and transition temperature tests’. ASTM STP 514, pp 164~191. 7.11 Akourri .O., Louah .M., Kifani .A. and Pluvinage.G. ,(2000).‘The effect of notch radius on fracture toughness J1c’. Engineering Fracture Mechanics N° 65, pp 491~505. 7.12 Sanz. G (1980). ‘Essai de mise au point d’une méthode quantitative de choix de qualités d’acier vis à vis du risque de rupture fragile’.Revue de Métallurgie CIT Juillet,(1980), pp 621~642. 7.13 Wallim. K.(1990). ‘Methodology for selecting Charpy Toughness. Criteria for thin high strength steels’. Part I, II, III Jernkontorets Forskning, Nr 4013/89 TO 40-05 –06-31 VTT manufacturing Technology Finland. 7.14 Eurocode 3.(1997). Part 2 ENV 1993 –2 Design of steel Structures, Steel Bridges. 7.15 Pluvinage.G., Krassowski.A.J, Krassiko.V.W. (1992.’Dynamic fracture toughness at crack initiation, propagation and arrest for two pipe-line steels’.Engineering Fracture Mechanics, vol 43, N°6, pp106~1084.

CHAPTER 8 NOTCH EFFECTS IN FATIGUE _____________________________________________________________ 8.1 Notch effects in fatigue and fatigue strength reduction factor Notch effects in fatigue are characterised by the fatigue life duration for the same stress amplitude being less, for a notched component than for a similar plain un-notched specimen, figure 8.1. Notch effects in fatigue can be quantified by way of the fatigue strength reduction factor kf kf

' V s N R , ' V n N R

(8.1 )

where 'Vs and 'Vn are respectively the stress range for a smooth and for a notched specimen at the same number of cycles to failure. Net stress range (MPa) 1000

100

Smooth Keyseat Notched

4

104

5

6

7

105 106 107 Number of cycles for fatigue life duration

Figure 8.1: Wöhler’s curves for smooth, key-seat and notched specimens (Construction steel E 360, rotating bending).

The fatigue strength reduction factor varies with the number of cycles to failure as can be seen in figure 8.2. For low cycle fatigue its value is near unity; for high cycle fatigue, it increases asymptotically to the elastic stress concentration factor value. 155

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G PLUVINAGE

The fatigue strength reduction factor is also a function of the notch radius. In the literature, numerous empirical formulae are available, allowing the fatigue strength reduction factor (at endurance limit only) to be derived from the elastic stress concentration factor. 8.2 Relation between fatigue strength reduction factor and stress concentration factor Numerous theoretical approaches make the assumption that the notch effect in fatigue is a result of the role of the maximum local stress. This is known as the ’Hot spot’ approach. Several comments can be made which undermine this assumption: i) the notch effect depends on the loading mode and specimen geometry for the same maximum stress or stress range values; ii) the position of maximum elasto-plastic stress is not at the notch tip and not connected with the point of fracture or fatigue initiation ; iii) the use of the elastic stress concentration coefficient kt results in over-conservative predictions vis a vis experimental results especially for low notch tip radii. However the use of the fatigue strength concentration factor is widely used and numerous solutions for various notch geometry and loading mode are available.

5

Fatigue strength reduction factor Notched

kt

Keyseat

4 3 kt

2

1 0

100

200

300

400

Net stress range Figure 8.2: Variation of kf as a function of number of cycles to fatigue failure for notched and key-seat specimens (Steel E 360).

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

157

For this reason considerable efforts has been conducted on the relationship between the elastic stress concentration factor and the fatigue strength reduction factor. In general, they refer to particular value of kf defined at endurance limit: kf

VD , V D, n

(8.2)

where VD is the fatigue limit on smooth specimens whilst VD,n is the fatigue limit on notched specimen. The relationship between the elastic stress concentration factor and the fatigue strength reduction factor kt may be classified into three categories depending on the assumptions used: (a) models using empirical relationships and based on the concept of an average stress over a given distance, (b) models based on the value of stress giving rise to a non-propagating short crack initiated from a notch, (c) models based on the localisation of fatigue damage in an effective volume. 8.2.1 MODELS USING EMPIRICAL RELATIONS These models [8.1~8.9] have established relationships between the elastic stress concentration factor and the fatigue strength reduction factor only on the level of the fatigue limit of the material VD. The mean stress at the notch tip for a distance called the ’effective distance‘ is regarded as equal to the fatigue limit. The stress distribution is considered to follow a 1/—r relationship in the earlier models. In more recent models the influence of loading mode and specimen geometry are introduced by way of the relative stress gradient F. These different models are presented in table 8.1. They are numerous and give different results, especially for low notch radii and need one or more empirical constants.For this reason the ’Hot spot‘ approach is not a convenient methodology. They incorporate one or two empirical constants depending on the mechanical characteristics of the material. These constants have been determined for several steels but not for modern-day ’clean low inclusion-content’ steels and were thus more sensitive towards fatigue. Therefore it is advisable not to use the constant values determined for these steels. ƒ The model proposed by Kuhn and Hardraht [8.1] assumes that fatigue failure occurs if the average stress range over a length Xef from the notch root is equal to the fatigue limit of the smooth specimen. The effective stress range at endurance limit may be expressed by the following equation : X ef 1 ' V ef , D (8.3) ³ V yy ( r )dr V D . X ef 0 ƒ The Peterson model [8.2] is based on the assumption that fatigue failure occurs when the stress, over a certain distance, is equal to or greater than the fatigue limit of a

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G PLUVINAGE

smooth specimen. Hence, the effective stress range at the endurance limit is equal to the stress value at a distance Xef from the notch tip: ' V ef , D ' V x X ef x ' V D . (8.4) ƒ Neuber's proposal [8.3] is very similar to that of Kuhn and Hardraht. He computed the average stress over a certain distance U' and considered a conventional notch radius U’= U+ LU', where U is the geometrical notch radius and L a constant depending on the specimen shape and loading mode. ƒ Switech [8.7] and Buch [8.8] have proposed that the effective distance corresponds to the distance where the stress distribution is equal to the endurance limit. ƒ According to Brand [8.13] it is possible to account fort the loading mode by use of the relative stress gradient at the notch tip: d V ij 1 1 d V ij lim . (8.5) F ij x o 0 V ij , max dx V ij dx x 0 Having analysed a considerable amount of fatigue data, obtained on smooth and notched steel specimens, the author suggests the following formulation for the fatigue limit: (8.6) Vˆ ij , Dn k ij V ij , Dn a log F ij  b ,

where VijDn is the fatigue limit of a notched specimen, Vˆ ijDn is the maximum elastic stress value for the applied (gross) stress equal to the fatigue limit stress, Fij the relative stress gradient, which depends on the specimen geometry, a and b are material coefficients. The constant values are determined for a non-failure probability of 90%. In order to predict the S-N curve evolution, Brand proposed the following equation:

Vˆ ij N R k ij V ij  N

R

a ' log N R  b' ,

(8.7)

where Vijn is the fatigue strength, a', b', are the parameters of Brand's formula relating the fatigue strength to the fatigue life duration, and Vˆ ijDn N r is the maximum elastic stress for a given applied stress. ƒ The influence of the relative stress gradient has also been taken into account by Switech [8.7] and by Wang and Zhao [8.11 ]. ƒ Ye and Wang [8.11] proposed a derivation of the fatigue strength reduction factor based on a modified Neuber’s rule,

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

kf

' V max ' H max , 'V N 'H N

kV k H

159

(8.8)

where ¨Vmax and ¨Hmax are respectively the maximum stress and the maximum strain ranges at the notch tip, ¨VN and ¨HN , the net stress and the net strain ranges, kV is the elasto-plastic stress concentration factor and kH the elasto-plastic strain concentration factor. Considering the fatigue strength reduction factor value is a function of the material properties, plastic deformation Hp at the notch tip and the elastic stress concentration factor kt, these authors proposed the following formula for kf: 1

kf

1 § 1  n' · 1 ¨ ¸. 1  'H el 'H pl © 1  n' ¹



,

(8.9)



where n’ is the cyclic hardening exponent and 'Hel and 'Hpl are respectively the elastic and plastic deformation ranges. The ratio between these two magnitudes introduces the material plastic damage and kt the geometry effect. Introducing into the above formula the equations of Basquin and Manson: c b ' H pl H ' f . 2 N R , ' H el V ' f . 2 N R 'H p 2Hcf (2 N R ) c , (8.10)

where 'Hel and 'Hpl are the elastic and plastic strain components, Vcf is the fatigue strength coefficient, b is the fatigue strength exponent, E is Young's modulus, Hcf the fatigue ductility coefficient and c the fatigue ductility exponent. Substituting in the equation (8.9) the relative formulae for the elastic and plastic components of strain, the mathematical expression of kf as a function of number of cycles to failure is : 1

kf

b  cº § 1  n' · ª 1 ¨ ¸. «1  V ' f E H ' f . 2 N R » 1 n '  © ¹¬ ¼



where V’f = Rm , H’f = Hf



b

 n'

1  5n'

,

c

1

1

1  5n'

.

,

(8.11)

(8.12)

These models have established relationships between the elastic stress concentration factor and the fatigue strength reduction factor only on the level of the fatigue limit of the material VD.

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The mean stress at the notch tip for a distance considered as the “effective distance” is regarded as equal to the fatigue limit. The stress distribution is related to 1/—r in some models, in the others the stress gradient F is used. 8.2.2 MODELS EMANATING FROM FRACTURE MECHANICS THEORY In these models previously discussed the stress amplitude at the endurance limit 'VD required for the non-propagation of a long crack in material and similar to the fatigue limit is compared with the stress amplitude 'Vth,n required for the non-propagation of a short crack initiated at the notch tip. The fatigue threshold 'Kth is equal to : ' K th

'V D S l 0 ,

(8.13)

where lo is a characteristic length. The effective fatigue threshold 'Keff is given by: ' K eff , th U th ' V D S l 0 ,

(8.14)

where Uth is the ratio of crack closure. The threshold for non-propagation of a short crack 'Keff,th ,with a length ath initiated from a notch is given by : ' K th

' V th, n S D  a th . F V a th W ,

D is a distance while F V a th W fatigue threshold 'Keff,th is :

(8.15)

is a geometric correction factor. The effective

' K eff , th, n U th, n ' V th, n S D  a th . F V a th W .

(8.16)

It is postulated that these two effective thresholds are identical and characteristic of the material: ' K eff , th ' K eff , th, n . (8.17) The fatigue strength reduction factor is defined by the ratio 'V D , kf 'V th, nc kf

U th, n S D  a th . F V a th W U th S l 0

(8.18) ,

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

U th, n . F V a th W

so that

U th

D  a th . l 0

161

(8.19)

This relationship is given by Ting and Laurence [8.15]. Author Kuhn et Hardraht [8.1] (1952) Peterson [8.2] (1959)

Formula kt 1

1

kf

1

S S \

.

Xc

U

k 1 1 t a' 1

kf

(8.18)

constants Xef material constant according to the yield stress

(8.19)

a’ = constant a’ = f (Vm)

(8.20)

a’ = constant a’ = f (Vm)

U

Neuber [8.3] (1968

1

kf

k t 1 a' 1

U

Heywood [ 8.4] (1952)

kf

kt

1

1 2

Stieler [8.5 ] (1954) Harris [8.6] (1961)

kf

kf

Switek.[8.7] and Bush [8.8] (1967) kf

Lukas and Klesnil [8.9] (1978)

kf

1

(8.21)

a'

U

kt 1  a' F

(8.22)

§ §  U ·· ¸ ¸ (8.23) 1  k t  1 .¨1  exp¨¨ ¸¸ ¨ a h © ¹¹ © § ¨1  2.1h ¨ U0  U © kt. A



a’ = constant a’ = f (Vm)



· ¸ ¸ ¹

§ U · ¸ 1  k t  1 .¨¨ ¸ © 4F V l 0 ¹ for U ” 4l0 kf = kt forU > 4l0

a’ = f(Re) and F is the relative stress gradient

ah = f(Rm) A and h are constants , U0 small distance

(8.24)

(8.25)

FVgeometrical factor l0 material constant

162

G PLUVINAGE

Topper and El Haddad [8.10] (1981)

kf

Wang and Zhao [8.11 ] (1984)

k t 1

1

1  4.5§¨ l c ·¸ © U¹ kt

kf

Boukharouba and al [8.12] (1995)

kf

(8.26)

0.88  A F b

kt





U 0  BU D X c  AU

(8.27)

(8.28)

lc: maximum length of a non propagating crack

A et b two constants of the material. U0 and Xc are small distances; A et B constants ; D slope.

Table 8.1 Formulae for the fatigue strength reduction factor.

However, if the short crack length ath is lower than the size of the notch plastic zone a*, Yu et al. [8.16] proposed the following relationship: kf

§a · F V ¨ th X ¸. © ¹

§ D  a th · ¨ ¸. ¨ l ¸ 0 ¹ ©

(8.20)

Yu et al. also obtained two expressions for kf in relation to kt by referring to the analysis for a non-propagating crack. kt for a /c = 1, (8.21) kf § ac · 1  1.4¨ ¸ © U¹ kf

kt

for a /c = 0,005;

(8.22)

1  3.5§¨ a c ·¸ © U¹

where ac is the critical length of the crack, a and c are the two half axes of the ellipse. 8.2.3 MODELS BASED ON THE STRESS FIELD INTENSITY In these models fatigue failure is assumed to be the result of the damage accumulation in a fatigue process volume. The accumulation of fatigue damage depends on the stress field intensity in the damaged zone. Referring to this concept, Sheppard [8.17] put forward the assumption that the fatigue strength of a notched specimen is related to the average Vmoy stress in this volume defined by:

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

kf

V moy V

M ,

163

(8.23)

where M is the domain of the stress field near the notch and V the applied stress. Yao et al. [8.18] defined the effective stress range by :

'V ef

& 1 ³ f V ij .I r dV , V ef :



(8.24)

where Vef is the effective volume where damage fatigue occurs, : is the region surrounding this volume; f is a weight function, and f(Vij) is the stress distribution function. If we use a non dimensional stress range ¨Vij / ¨VN where ¨VN is the net stress, the effective stress range is given by: 'V N & (8.25) 'V ef ³ f V ij .I r dV , V ef :



The fatigue strength reduction factor can be obtained from: ' V N R & 1 kf ³ f V ij .I r dV , ' V ef N R V ef :



(8.26)

In many cases the volume of the fatigue process volume is regarded as a revolution solid with a height equal to the thickness of the structure, which leads to a bidimensional problem : kf

& 1 ³ f V ij .I r ds , SD



(8.27)

where D is the surface of the process volume and S its area. An additional simplification may be provided if we consider that the fatigue process volume is cylindrical with a diameter equal to Xef. In this case : & 1 (8.28) kf ³ f V ij .I r dx , X ef L





where L plays the part of D. If we consider that f V ij equals V ij and I (x) = 1. kf

X ef ³ V ij dx . X ef 0 1

(8.29)

164

G PLUVINAGE

8.3

Volumetric approach

8.3.1 PRINCIPLE OF THE VOLUMETRIC METHOD The volumetric approach is an alternative and innovative way of modelling the fatigue failure process emanating from notches. The assumption made in this approach is that the fatigue failure needs a physical volume to occur. Its extent from the notch tip is called the effective distance. This approach was first introduced for an elastic stress distribution [ 8.19]. Since then, it has been subject to further development powing the increasing knowledge concerning scale and loading mode effects on fatigue. In [8.20] the method was applied adopting an elasto-plastic stress distribution calculated using a finite element method and applying the cyclic behaviour in order to take into account the plastic and damage relaxation. The effective stress according to the volumetric approach was first determined as the stress value corresponding to the stress distribution for the effective distance. It is now defined as the average of the weighted stresses in this volume. This weighted stress depends on the relative stress gradient in order to account for the loading mode and scale effects in fatigue. The fatigue life duration determined by traditional methods requires the value of the fatigue strength reduction factor, which is a function of the stress concentration factor kt Figure 8.3. shows two kinds of specimen geometry having the same stress concentration factor value kt = 2.72. Based on the traditional approach, these geometries should have the same fatigue life duration. In fact, the fatigue life duration of each specimen type is different from the other. It should be noted that the stress distribution near the notch tip is different and U2 consequently the first derivative of the stress distribution function is also different. This fact can explain the difference in fatigue strengths of the two specimens.

db

U1

dV/dx da a) da= 8 mm, Da= 10 mm

dV/dx Db

Da b) db= 7,2 mm, Db= 15 mm

Figure 8.3: Different stress distributions for two specimens with the same stress concentration factor k t (kt = 2.72), notch radii U1§ U = 0.3 mm).

The relative stress gradient is defined as the ratio between the first derivative of the stress distribution function and the value of the stress at the point.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

1

F

V ij

.

d V ij dr

,

165

(8.30)

where F is the relative stress gradient which plays an important role in the fatigue process as mentioned previously by Brandt [8.21] and Siebel and Stieler [8.22]. 8.3.2 EFFECTIVE DISTANCE The fatigue process volume is assumed to be cylindrical with a diameter denoted Xef. This assumption is based on an analogy with the notch plastic zone which is practically cylindrical. The fatigue process volume is the high stressed region ahead of the crack or notch tip. If we considered a typical elastoplastic stress distribution at a notch tip (figure 8.4) exhibiting 4 classical regions, it can be seen that it is not easy to defined precisely the end of this high stressed region in the upper part of the distribution. Two particular points are visible on the distribution: i) the point where the stress is at maximum; ii)the limit between region II and region III.

Stress range( MPa)

V 5 F I

0

500

I

0 0

II

III

IV

0.5

1.0 Distance (mm)

Relative stress gradient (mm-1)

1000

-4

Figure 8.4: Stress distribution, relative stress gradient, and weight function versus the distance; notched specimen in carbon steel submitted to rotating bending; notch radius U = 0.2mm.

The choice of the value of the effective distance was made by trial and error method. By definition, the effective stress leads to the same lifetime as that given when testing smooth specimens of the same material. Verifications on different materials and specimen geometries [8.23] have shown that the limit between region II and III can be considered as the effective distance with a high degree of confidence.

166

G PLUVINAGE

We can also see that this point is an inflexion point and corresponds to the minimum value of the relative stress gradient. An inflexion point leads to the following conditions for the second derivative: 'V " 0 for X<Xef and 'V "! 0 for X>Xef , (8.31)

If we consider the stress distribution in zone III: CD CD .D  1 C 'V , 'V ' , 'V " , D  1 D x xD  2 x

(8.32)

and determine the first derivative of the relative gradient from : dF 'V " 'V ' 2 'V "   F 2 ., (8.33) dx 'V ' 'V ' 'V We find that: (a) in region III, the first derivative of the gradient is always positive (D is positive) dF D ! 0 for x > Xef, (8.34) dx x 2



(b) in region II, the first derivative of the gradient is always negative : dF  0 for x < Xef. (8.35) dx Consequently the minimum of the relative stress gradient occurs for a distance to the notch tip equal to the effective distance which corresponds precisely to the beginning of zone III in figure 8.4. This leads to a graphical procedure allowing an exact determination of the effective distance which corresponds to the minimum of the relative stress gradient. 8.3.3 EFFECTIVE STRESS The volumetric method assumes that all the stress points in the process volume play a role in the fatigue process. The weight of each point is different and is influenced by two parameters : (i) the distance between a stressed point and the notch tip. When this distance increases, its influence on the fatigue process decreases; (ii) the relative stress gradient, which takes into account the influence of the loading mode, geometry and scale effect as previously seen. A higher stress gradient (absolute value) increases the influence of the stress point. The role of the stress intensity, the distance and the relative stress gradient are combined in a ’weighted stress‘ V*ij. This weighed stress is defined as: (8.36) V *ij V ij .I r , F ,

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

167

where I(x,F) is a weight function which depends upon the distance r and the relative stress gradient F. The weight function is assumed to be at the notch tip and at the point of maximum stress equal to unity. For this reason, the choice of the weight function has the following form: (8.37) I r , F 1  rF . The effective stress Vef is defined as the average value of the weighted stress in the fatigue process volume: 1 . ( 8.38) V ³³³ V * dV ef

V ef V ef

ij

During fatigue propagation, the crack path is always perpendicular to the maximum principal stress. In tension or bending, this stress is conventionally denoted Vyy. We can write in bi-dimensional case: 1 . (8.39) V V * dr ef

³ X ef X ef

ij

8.3.4 EXAMPLE Fatigue tests have been carried out under rotating bending loading using steel specimens having different surface geometries namely: a) smooth specimens to obtain the reference curve, b) specimens with keyseats. Two types of key-seat have been tested: Sled Runner and semi-circular end. The geometry of these two types of key-seat is presented in figure 8.5.The experimental results of these tests is presented in figure. 8.6 as Wöhler’s curves for smooth specimens and 3 types of key-seats (semi-circular and Sled Runner key-seat with two radii R = 5 and 7.5 mm). From these results we can see that the introduction of a key-seat causes the reduction in the fatigue lifetime. However for this sled runner key-seat, the radius R has a very important role in fatigue life duration. In order to apply the volumetric approach to these experimental results, finite element computing of the stress distribution at the tip of key-seats have been accomplished using the ‘CASTEM 2000’ finite element software and the following parameters: (cyclic hardening coefficient K’ = 920 MPa and cyclic hardening exponent n’ = 0.023 ).

168

G PLUVINAGE

R

a) semi-circular end key-seat

b) Sled Runner key-seat

Figure 8.5 ; Geometry configuration of different key seats.

Stress range (MPa)

1000

R

Smooth Semi-circular End Seld Runner R=5 Seld Runner R=7,5 100 1000

10000

100000

1000000

10000000

100000000

Number of cycles for fatigue life duration Figure 8.6 : Wöhler curve for smooth specimens and 3 types of key-seats (semi-circular end Sled Runner key-seat with two radii R = 5 and 7.5 mm).

The effective stress may then been calculated according the procedure describe in 8.3. and plotted versus the experimental number of cycles to failure. The results fit with a good level of confidence to the Wöhler curve relative to the smooth specimen reference curve (figure 8.7). The volumetric method is able to predict the fatigue life duration for any notched structure. To apply this method you need the following data :

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

169

í the fatigue reference curve of material determined on smooth specimen, í the cyclic strain stress curve of the material. Effective stress obtained by volumetric method reported on fatigue reference curve gives directly the fatigue life duration (Figure 8.7). Safety factor can be introduced to get admissible stress which is the ratio of the effective stress by safety factor (a value of 2 can be used).

Stress range (MPa) 1000

Reference curve Semi-circular End Seld Runner R=5 Seld Runner R=7,5

100 10000

100000 Nr

1000000

Number of cycles for fatigue life duration Figure 8.7: Application of the Volumetric Method to fatigue tests using specimens with key-seats; comparison of experimental results with fatigue reference curve.

It is important to mention here that, it is not possible to apply other fatigue models to specimens with key-seats. This is because almost all of these models refer to the notch radius, which cannot be determined in the case of the specimens with key-seats. The advantages of the volumetric method include the possibility of predicting fatigue life duration for any loading case using notched geometry structures, the absence of empirical and doubtful coefficients used in traditional methods and the opportunity to obtain fast and economical results using a Finite Element method.

170

G PLUVINAGE

8.4 Influence of loading mode 8.4.1 INFLUENCE OF LOADING MODE ON FATIGUE LIFE DURATION OF SMOOTH SPECIMENS Loading mode also has an important effect on the fatigue life duration of notched specimens. In order to study this influence tests have been performed on notched specimens made in carbon steel (yield stress Re = 312 MPa and ultimate stress Rm = 500 MPa). The geometry of these specimens is presented in Figure 8 .8. 2T

U



I7,62

+0,05 - 0,05



Figure 8.8: Notched specimen for push-pull and torsion fatigue tests and notch geometry; (Dimensions in mm).

SíN curves (stress amplitude versus fatigue life duration) for smooth specimens subject to alternate torsion, alternate tension-compression and rotating bending loading, are presented in figure 8.9. . 1000

Stress range Torsion Tension - compression Rotating bending Equivalent stress in torsion

100 103

104

105

106

107

108

Number of cycles for fatigue life duration Figure 8.9: fatigue curves for smooth specimens under different loading modes.

This figure shows loading mode has a strong influence on smooth specimen fatigue life duration for the same stress amplitude.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

Tensioncompression Rotative bending Torsion

171

b

V’f (MPa) 470

0.05 0.08 0.04

887 337

Table 8.2: Basquin’s coefficient and Basquin’s exponent of the fatigue curve under different loading modes

In this figure data from rotating bending and alternated tension-compression tests are presented as normal stress amplitude versus number of cycles to failure. The data from torsion tests are presented into two ways: shear stress amplitude versus number of cycles the curve and equivalent normal stress according to Von Mises criterion Analysis the test results allows Basquin’s law parameters to be determined. The general Basquin’s law has the following general form: b, m 'V V f , m  N R (8.40) where ¨V V’f,m and bm are respectively the applied stress range, Basquin’s coefficient and Basquin’s exponent of the fatigue curve under one of the above mentioned loading modes (m = T, t ,b tension, torsion or bending). The results of this analysis are presented in table 8.2. 8.4.2 INFLUENCE OF LOADING MODE ON FATIGUE LIFE DURATION OF NOTCHED SPECIMENS A plot of the fatigue test results for smooth specimens (reference curve) and notched specimens with notch radii U= 0.4mm and U= 0.2mm, subjected to rotating bending loading, tension–compression and torsion, is presented in figure 8.10. Stress range (MPa) 1000

Rorsion Tension-compression Rotating bending Von Mises stress for torsion

500

100 103

104

105

106

107

108

Number of cycles for fatigue life duration

Figure 8.10: Stress range versus fatigue life duration for fatigue tests using smooth and notched specimens under rotating bending, tension –compression and torsion.

172

G PLUVINAGE

The coefficient and exponent of Basquin’s law together with correlation coefficients for each fatigue curve are reported in Table 8.3 The same procedure is followed for fatigue tests conducted under the two other loading modes. Specimen type b R2 V cf (MPa) Smooth (reference) 887.59 -0.0871 0.8901 -0.154 0.8932 Notched U= 0.4 1284.3 mm -0.1727 0.9131 NotchedU= 0.2 1449.4 mm Table 8.4 Basquin’s law parameters and correlation coefficients for rotating bending fatigue curves shown in figure 8.10.

Fatigue strength reduction factor 2

Bending

Carbon Steel Re = 312 Mpa

Tension 2 Torsion 11 0 104

105

106

Figure 8.11: Variation of the fatigue strength reduction factor versus the number of cycles to failure for 3 modes of loading (tensioncompression, rotating bending and torsion); specimen notch radius 0.2 mm.

8.4.3 INFLUENCE OF LOADING MODE ON FATIGUE STRENGTH REDUCTION FACTOR The effect of different loading modes on the fatigue behaviour can be seen from both the SíN curves and the Basquin’s law parameters. In order to see the influence of the loading mode, variation of the fatigue strength reduction factor versus the number of cycles to failure has been plotted. The fatigue strength reduction factor is defined as

kf

' V  N R ' V  N R

smooth .

(8.41)

notched From this figure we can see that the fatigue strength reduction factor increases linearly with the logarithm of the number of cycles for life duration. Values vary from near unit

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

173

value for low cycle fatigue for each type of loading and increase more rapidly in bending than for tension. A small increases occurs in torsion (kf = 1.16 for 10,000cycles and kf = 1.47 for 10 million of cycles). The fatigue strength reduction factor is more important in bending and increases more rapidly with the number of cycles than for torsion. 1000 Tension -compression reference curve

100 Vol. (no gradient)

Computed Vol. (with gradient) effective stress

10

S-N curve U = 0,4mm

Ye & Wang Brand

104

105

106

107

108

Number of cycles for fatigue life duration Figure 8.12.a : Experimental Wöhler curves on notch and smooth specimens presented along with computed data determined from the volumetric method tension compression.

The principle of the volumetric method has been applied to tests performed in tensioncompression figure 8.12.a and torsion figure 8.12.b on notched specimens (notch radius 0.4mm) manufactured from a low strength steel (yield stress Re = 312 MPa). Experimental Wöhler curves on notch and smooth specimens are presented with the computing data (full square dots). We can note the good agreement between the prediction of the volumetric method and the fatigue reference curve (smooth specimens curve). 8.4.4 INFLUENCE OF HYDROSTATIC PRESSURE ON FATIGUE LIFE DURATION OF NOTCHED SPECIMENS 8.4.4.1 The hydrostatic pressure The hydrostatic pressure Vh is defined as the first invariant of the stress tensor. If VI, V2, V3 are the three principal stress, the hydrostatic pressure is given by: V1V 2 V 3 , (8.42) Vh 3 In the case of torsion, figure 8.13(a) V2 = 0 ; V1 =-V3 and Vh = 0; In the case of traction, figure 8.13 (b) V2 = V3 = 0; and Vh = V1 /3; In the case of tension + torsion, figure 8.13 (c) and Vh = (V1 +V2 + V3 )/3.

174

G PLUVINAGE

Shear stress range (MPa) 1000 CARBON STEEL Re = 312 MPa

500

100

102

103 104 105 106 107 108 Number of cyles for fatigue life duration

Figure 8.12 b: Experimental Wöhler curves on notch and smooth specimens presented along with computed data determined from the volumetric method, torsion.

W

V3

W

Torsion V

V2

V1

a) torsion Vh = 0

V2 V3

Tension W

Tension + Torsion

V2

V1

V V1

b) tension Vh = V1 /3

V

V3

c Vh = (V1 +V2 + V3 )/3

Figure 8.13 : Representation of three-dimensional state of stress using Mohr circles for 3 stress states.

8.4.4.2 Influence of hydrostatic pressure on fatigue life duration Hydrostatic pressure has been considered by several authors as an effective parameter for fatigue resistance. Several approaches are possible and the fatigue criterion can be a combination of the second stress invariant J2 and the hydrostatic pressure (Table 8.5) ;. Authors Sines [8.24]

Crossland [8.25]

Formula

J 2 N V h d O J 2  N V h, max d O Table 8.5 Relationship of Sines [8.24] and Crossland [8.25].

(8.43)

(8.44)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

175

Dang Van [8.27] has considered a combination of microscopic shear stress and hydrostatic pressure. In formulas (8.43) to (8.44), N and O are material constant, V h, max the maximum value of hydrostatic pressure and V h mean value According to Dang Van, fatigue crack initiation occurs in critical zones with stress concentrations or near plastic sliding in grains which are favourably oriented with respect to external loading. Analysing the local stresses and transferring this analysis to a macroscopic scale, Dang Van defined a fatigue initiation criterion at a point and at a time which satisfies the following condition: W  A DV V h d B DV (8.45) where ADV and BDV are material constants. The basic mechanism for fatigue crack initiation is the maximum shearing stress which occurs on the most favourably oriented crystallographic plane. The maximum shear stress and the plane of maximum shear stress have to be determined in order to apply this criterion. The negative influence of hydrostatic pressure increases linearly. The two constants are determined for two particular states of stress : torsion where the fatigue limit is WD and hydrostatic pressure is equal to zero and alternate tension where the hydrostatic pressure is VD and the fatigue limit VD : §W 1· W  ¨¨ D  ¸¸.V h W D . ©V D 2¹

(8.46)

The Dang Van model is the basis of Flavenot and Skalli’s [8.28] critical layer criterion. However, instead of computing the maximum shear stress and hydrostatic pressure at the surface, they proposed to computing the average values over a‘ critical layer‘ which has the same meaning as that of the effective distance. Examining a large range of experimental data on steel notched specimens, they found that all the data fitted the fatigue endurance curve W = f(Vh) determined for smooth specimens. The best value for the critical layer is determined by a trial and error method and its value is of the order of the material characteristic, e.g. grain size. In the case of fatigue under combined tension and torsion the relation between shear stress and hydrostatic pressure is given by: 2W  V D W  3. D .V H W D (8.47) VD An example of the Dang Van diagram is given for a steel (Re = 312 MPa) in figure 8.14. 8.4.4.3 Example for combined loading tension and torsion Influence of hydrostatic pressure on fatigue life duration is examined for combined tension + torsion tests. The ratio between the shear and normal gross stress is 2. The geometry of the smooth and notched specimens used is described in figure 8.8.

176

G PLUVINAGE

The material used is a low carbon steel with yield stress Re = 312 MPa and ultimate stress Rm = 500 MPa. Shear stress range (MPa) 200

150

0.5 1

100 Carbon steel Re = 312 Mpa; Rm = 500 MPa WD = 124 MPa ;VD = 194 Mpa

50

Hydrostatique pressure Vh (MPa)

0 0

50

100

150

200

Figure 8.14: Example of the Dang Van diagram for a carbon steel .

Hydrostatic pressure (MPa)

F (mm-1) V

Smooth specimen

1

Vh V1 100

Ft0.1

W F Xef

10

p W

0.1

1

Distance (mm)

0

-0.1 10

Figure 8.15.a: Variation of maximum principal stress V1, maximum shear stress W, hydrostatic pressure Vh and relative stress gradient F in a bilogarithmic diagram.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

177

The notched specimens are of two types with different notch radius U = 0.2 and U = 0.4 mm. Basquin law coefficient and exponent for the Wöhler curves have been determined and reported in table 8.8 b

R2

smooth

V’f (MPa) 718

-0.066

U = 0.2 mm U = 0.4 mm

911 1035

-0.1335 -0.1526

0.934 5 0.813 0.897 4

Specimen

Table 8.8 Basquin law coefficient and exponent for the Wöhler curves.

600

Hydrostatic pressure + Wand V1 (MPa) Notched specimen

F(mm-1) V

1

p W

400

F

t

2 1 0

200 -1 0

0

X x ef

1

2 Distance (mm)

Figure 8.15.b: Variation of maximum principal stress V1, maximum shear stress W, hydrostatic pressure Vh and relative stress gradient F in a bilogarithmic diagram.

Using Finite Element Method and assuming elastoplastic material behaviour, maximum principal stress V1, maximum shear stress W, hydrostatic pressure Vh and relative stress gradient F were computed. An example of such computations is given in figure 8.15 for (a) a smooth specimen loaded in tension and (b) a notched specimen (U = 0.4mm) loaded with combined tension + torsion. From figure 8.15, we note a higher relative stress gradient for torsion loading. In order to take into account the influence of the stress gradient, the effective maximum shear stress and hydrostatic pressure are computed; These quantities are defined as the average values of maximum shear stress and hydrostatic pressure over the effective distance which is defined as the distance of minimum relative stress gradient.

178

G PLUVINAGE

300 Effective maximal shear stress (MPa) Tension + torsion Vg/Wg =2 200

100

0

1E+3 cycles 1E+4 cycles 1E+5 cycles 1E+6 cycles

Carbon steel Re = 312 MPa 50

0

100

150

250

200

Effective hydrostatique pressure (MPa) Figure 8.16: Evolution of effective maximal shear stress versus hydrostatic pressure.

8.4.4.4 An elliptical criterion for the influence hydrostatic pressure of fatigue life It is assumed that in the Dang Van and Flavenot Skalli’s model, the influence of hydrostatic pressure is linear; plotting the effective hydrostatic pressure Vh,ef for the previous experimental results and for different life duration, it is noted that the assumption of linear dependence is not used (Figure 8.16). We can see that the size parameters of such elliptical representation is not constant but depends on fatigue life duration (Figure 8.17). Axis range (MPa) 1000

W Axes horizontaux CARBON STEEL Axes=verticaux Re 310 MPa

Vh R2 = 0.9715

R2 = 0.9916

100

102

103 104 105 106 107 Number of cycles for fatigue life duration

Figure 8.17 : variation of axis size

of elliptical criterion versus number of cycles.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

179

8.5 Notch effects in low cycle fatigue Although there is an important plastic relaxation at the notch tip, fatigue notch effect still exists in low cycle fatigue. This can be seen in figure 8.17, the fatigue strength reduction factor for a value of cycles to failure less than 100 000 is never equal to one but varies in the range [1.1-1.4]. A typical problem is that of low cycle fatigue crack initiation at stress concentration at a nozzle in a pressure vessel. Notch effects in low cycle fatigue have received little attention according to the literature. The actual trend is to treat this problem using an energy approach for low cycle fatigue. Young’s modulus E (GPa)

203

Yield stress Re (MPa)

515

Cyclic strain hardening exponent n’

0,15

Cyclic strain hardening coefficient K’(MPa)

1219

Table 8.9 mechanical properties of 35 NCD 16 steel :

Evidence of notch effects in low cycle fatigue can be illustrated from experimental results obtained for tests on axi-symmetric specimens loaded in tension-compression. The notched specimens have a notch radius of U = 0.4 and 1.2 mm. The steel studied is a 35 NCD 16 steel (French standard) with the following mechanical properties: Experimental results are presented using the traditional Coffin [8.29 ] and Basquin’s laws [8.30]. c ' H pl H ' f . 2 N R , (8.48)

' H el

V'f E

b . 2 N R ,

(8.49)

where 'Hpl is the plastic strain range and 'Hel is the elastic strain range. V’f and H’f are the fatigue strength and ductility to fatigue, b and c are exponents. The Manson-Coffin‘s law [8.31] is expressed by: V' f b c ' H pl . 2 N R  H ' f . 2 N R , E

(8.50)

where ¨Ht is the total strain range. The values of these parameters are presented in Table 8.10 and an example of each of these 3 laws is presented in figure 8.18.

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G PLUVINAGE

V' f E b H'f c

smooth 0.57

U = 0.4 mm 0.70

U = 1.2 mm 0.33

- 0.10 5.15

- 0.17 27.54

- 0.08 16.94

- 0,37

- 0.95

- 0.88

Table 8.10 : Values of the parameters of the Manson-Coffin‘s law for a 35NCD16 Steel.

The fatigue resistance curve can be expressed in another way using the relationship between the strain energy density range ¨W* and the number of cycle to failure: 'W *

a A. N R ,

(8.51)

where A and a are material constants. The strain energy density range ¨W*p which represents the area of the hysteresis loop can be computed from the stress and the plastic strain range using the Halford relationship [8.32]:

§ 1 − n' · ∆ε p = ¨ ¸∆σ .∆ε , © 1 + n' ¹

(8.52)

where n’ is the cyclic strain hardening exponent. Using the cyclic stress-strain range relationship: 'V 1 n' (8.53) ' H pl K' We get the plastic strain energy density versus the stress range 'V.)



Strain range 'Hp/2 35 NCD 16 STEEL 10-1

elastic plastic total

10-2

10-3 10-4

103 104 105 102 10 Number of cycles for fatigue life duration

Figure 8.18: MansoníCoffin, Manson and Basquin ‘s curves for a notched specimen (U = 1.2 mm) for a 35NCD16 Steel.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

181

The total strain energy density range is given by the sum of the elastic part ¨W*el and the plastic part ¨W*pl. § 1  n' ·  1 n' (8.54) ' W *pl ¨ . 'V 1  n' n' . ¸. K ' © 1  n' ¹ The same relationship is used for notched specimens although the strain energy density range is not constant in the specimen volume. This computation gives an effective strain energy density range which is assumed constant in the effective volume.

A a

smooth 4385 - 0.29

U = 0.4 mm 2211 - 0.31

U = 1.2 mm 1977 - 0.28

Table 8.11: Values of parameters of the fatigue resistance curve expressed in terms of strain energy density.

Strain energy density range 'W* (MJ/m3) 104

Steel 35 NCD16

smooth U=0.4 mm

103

102

10 102

103

104

105

Number of cycles for fatigue life duration Figure 8.19: Strain energy density range versus the number of cycles to failure for smooth and notched specimens for a 35NCD16 Steel.

This value is used to define the fatigue strain energy density concentration factor. Experimental results are presented in figure 8.19 and Table 8.11.It can be seen that the three relationships have the same value of the coefficient ’a‘ (average value ’a‘ = 0.30) and the curves relative to the notched specimens are similar to the smooth one. The strain energy density at the notch tip has been computed using a finite element method. Calculations have been made using the cyclic stress strain curve. The strain energy density range is plotted versus the distance from the notch tip in a bi-logarithmic graph. A typical example of such a curve is given in figure 8.20. The strain energy density distribution diagram can be divided into 3 parts: zone I where the strain energy density range is practically constant and equal to the maximum range value

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G PLUVINAGE

§ 'W *max · ¨ ¸ ¨ 'W * ¸ N ¹ ©

 K W *

2

;

 

zone III where the strain energy density range exhibits a power dependence with the non dimensional distance (where C and D’ are constants, D’ > 1) : D ' § 'W *max · ¸ C r ¨ ; (8.56) ¨ 'W * ¸ b N ¹ ©



zone II intermediate zone between zones I and III. The distribution in zone III can be assimilated to a “pseudo strain energy density singularity ”. This distribution can be considered only for a distance greater than Xef with the following formula : 'K U , W * for r> Xef. (8.57) 'W * D'  2Sr Xef is also called the effective distance and represents the diameter of the fatigue process zone ,being assumed cylindrical. 1000

Strain energy density (J/m3) STEEL 35 NCD 16 U = 0.4 mm

100

10 1

5

10 Distance (mm)

Figure 8.20: Strain energy density distribution from the notch tip presented in a bi-logarithmic graph. for a 35NCD16 Steel (U = 0.4 mm).

The effective strain energy density range is the average value of the distribution over the distance Xef. According to our previous definition the value of the effective strain energy density range 'W*ef leads to the same number of cycles for the same value of effective strain energy density range 'W* applied to a smooth specimen.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

183

Figure 8.21: Comparison between the computed and experimental number of cycles to failure based on the effective strain energy density for a 35NCD16 Steel.

The accuracy of this definition can be checked by comparison of the number of cycles to failure obtained experimentally and those obtained using the value of the effective strain energy density range. Such a comparison indicates that this definition gives relatively satisfactory results (figure 8.21).

Fatigue strength reduction factor kf 2

1 ,7 5 r o =0.4 1 ,2mm mm U= 1 ,5

1 ,2 5

r o = 1.2 0 ,4 mm mm U=

1

2500 5000 7500 0 Number of cycles for fatigue life duration Figure 8.22 : Variation of the fatigue (strain energy density) concentration factor with the number of cycles to failure for a 35NCD16 Steel.

The fatigue (strain energy density) concentration factor kf,W* is defined in a similar manner that for high cycle fatigue

184

G PLUVINAGE

k f ,W * 2

§ 'W *s · ¨ ¸ (8.58) ¨ 'W * ¸ . n¹ © where 'W*s and 'W*n are respectively the strain energy density range for smooth and notched specimens leading to the same number of cycles to failure.

Variation of the fatigue strength reduction factor defined from strain energy density is plotted versus the number of cycles for two notch radii and presented in figure 8.22. We can see that the fatigue strain energy density factor is practically constant with the number of cycles, i.e. 1.62 for U= 0.4 mm and 1.42 for U = 1.2 mm.

8.6 Conclusion Fatigue life duration for notched structures can be predicted from the fatigue reference curve obtained on smooth specimens using two approaches: Hot Spot approach and volumetric method. The Hot Spot approach requires the use of fatigue strength reduction factor which may be obtained from several empirical relationships which can include two or more material constants and the relative stress gradient. The volumetric method uses the assumption that the fatigue process requires a physical volume on which an effective stress range operates. This effective stress range is an average value of the weighted stress distribution inside the fatigue process volume. This effective stress range is sensitive to hydrostatic pressure and consequently to loading mode. Notch effects still exist in low cycle fatigue and can be described using the fatigue strain energy density factor.

REFERENCES 8.1 Kuhn.P. and Hardraht.H.F.(1952).’An engineering method for estimating notch size effect in fatigue tests on steel’, NASA, Technical Note, No 2805. 8.2 Peterson.R.E. (1959) ’Notch sensitivity’, in G. Sines and J. L. Waissman (eds.), Metal Fatigue, McGraw Hill, New York, pp 293~306. 8.3 Neuber. H. (1968). ’Theoretical determination of fatigue strength and stress concentration’, Air Force Material Laboratory, Report AFML-TR-68-20. 8.4 Heywood.R.B .1952). ’ Designing by photo-elasticity’. Chapman and Hall, London. 8.5 Stieler. M, (1954).’Untersuchung ueber die Dauershwingfestigkeit metallischer Bauteile bei Raum Temperatur’. Dissertation Techn. Hoschule, Stuttgart. 8.6 Harris.W.J .(1961).’Metallic Fatigue’ .International Series of Monographs in Aeronautics and Astronautics, Pergamon Press. 8.7 Switech.W(1967)..’ Effect of notch parameters and stress concentration on the fatigue strength of steel’. (in german), Ifl, Mitt. N° 11. 8.8 Buch. A. (1974).’Analytical approach to size and notch size effects in fatigue of material specimens’.Materials Science and Engineering, Vol. 15, pp 75~85. 8.9 Lukas. P and Klesnil.M . (1978). ’ Fatigue limit of notched bodies’. Material Science and Engineering, 34, , pp 61~66. 8.10 Topper.T.H. and El Haddad.M.H . (1981).’Fatigue strength prediction of notches based on fracture thresholds. 1st Int. Conf., Stockholm, vol.2, EMAS, Warley, UK.

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8.11 Wang.Z. and Zhao.S. (1992).’Fatigue design’, Mechanical Industry Publisher, (In Chinese). 8.12 Boukharouba. T, Tamine.T, Nui.L, Chehimi.C,and Pluvinage G. (1995).’ The use of notch stress intensity factor as a fatigue crack initiation parameter’.Engng. Fract. Mech, Vol N°. 3 , pp 503~512. 8.13 Brand, A. and Sutterlin. R.(1980).Calcul des pièces à la fatigue. Méthode du gradient, CETIM, France. 8.14 Ye.D.Y and Wang. D. J. (1996).’A new approach to the prediction of fatigue notch reduction factor kf ’, International Journal of Fatigue, Vol 18, N° 2, ,pp 105~109. 8.15 Ting, J. C. and Lawrence Jr, F. V. (1993). ’Fatigue and Fracture of Engineering Materials and Structures, 16, pp93. 8.16 Yu, M. T., Du Quesnay. D. L. and Tipper. T. H.). (1993). International Journal Fatigue, 15, p109. 8.17 Sheppard, S. D. (1989). in ’Failure Prevention and Reliability’, N°89, ASME, New York, pp. 119~27. 8.18 Yao, W. “International Journal of Fatigue“, 15, (1993), pp 243. 8.19. Niu, L. Chehimi. C and Pluvinage.G.(1994).’Stress field near a large blunted tip V-notch and application of the concept of the critical notch stress intensity factor (NSIF) to the fracture toughness of very brittle materials’. Engineering Fracture Mechanics ,Vol. 49, No. 3, , pp.325~335. 8.20 Qylafku.G., Azari.Z., Kadi.N., Gjonaj.M and Pluvinage.G. (1999). ‘Application of a new model proposal for fatigue life prediction on notches and key-seats’, International Journal of Fatigue 21, pp753~760. 8.21Brand.A., Flavenot. J. F., Gregoire.R. and Tournier.C. (1991). ‘Données technologiques sur la fatigue’, CETIM, 2d edition, France. 8.22. Siebel. E. and Stieler.M.(1955).‚Non-Uniform Streses Distribution during Fatigue Loading’. VDI-Z, 97, 8.23 Pluvinage.G. (1997). ‘Notch effect in high cycle fatigue’. Advances in Fracture Research, ICF9, Fatigue of Metallic and Non-metallic Materials and Structures, 3, Pergamon Press, Sidney, pp.1239~1250. 8.24 Sines.G.(1995).’Behaviour of metals under combined repeated stresses with superimposed static stresses’. National Advisory Committee for Aeronautics Technology, Note3495, Nov. 8.25 Crossland.B.(1949) ’Effect of large hydrostatic pressure on the torsional fatigue of an alloy steel’. Proceedings of the International Conference of Metals, Institution of Mechanical Engineers,London. 8.26 Kakuno.H. and Kawada.Y. (1979).’ A new criterion for fatigue strength of a round bar subjected to combined static and repeated bending and torsion”, Fatigue of Engineering Materials and Structures, Vol 2. 8.27 Dang Van.K, (1997).’High cycle fatigue analysis in mechanical engineering’. Advances in fracture research ICF9,Vol 3, pp 1225~1237. 8.28 Flavenot.J.F and Skalli.N. (1982). ’ L’épaisseur de couche critique ; une nouvelle approche du calcul en fatigue des structures soumises à des sollicitations multiaxiales’. Rapport CETIM 12 G 254, Septembre, pp15~25. 8.29 Coffin. L. F.(1969). ’Manual on low cycle fatigue testing’, ASTM STP 465. 8.30 Basquin O. H. (1969). ’The experimental law of endurance test’.ASTM, Vol. 10, pp. 625. 8.31 Manson. S. S, (1965). ’Fatigue : a complexe subject. Some simple approximation‘Experimental Mechanics, 8.32 Halford. G. R. (1966).’The energy required for fatigue’. Journal of Materials, Vol. 1, (March) , pp. 3~17. 8.33 Morrow. J. D and Tuller. F. R. (1965). ’Low cycle evaluation of Inconel and Waspaloy", ASME, Journal of Basic Engineering, vol 27, N°2, pp 275~289.

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CHAPTER 9 ROLE OF STRESS CONCENTRATION ON FATIGUE OF WELDED JOINTS ___________________________________________________________________ 9. 1 Introduction The fatigue life of welded joints is markedly affected by the existence, at the weld toe, of a stress concentration; the local stresses are associated with the welded joint geometry and/or the presence of a pre-existing defect. The existence of a stress concentration is a deciding factor in the behaviour of a structure under fatigue loading. Nikei et al [9.1] studied the variation in fatigue strength (at 5*105 cycles) for different types of joint according to the stress concentration factor. The results presented in figure 9.1 show a significant decrease in fatigue strength according to kt. The coefficient of the elastic stress concentration kt of a welded joint depends on geometry and loading. Some analytic formula can be found in the literature and allow the user to calculate the stress concentration factor; loading effects on the value of kt. are taken into account. 300

Stress range (MPa)

Butt joints

cruciform cruciform K 200 NR= 105 cycles

100 1

Stress concentration factor kt

100

Figure 9.1 : Relation between mean stress for the fatigue strength at 5* 105 cycles and the logarithm of the stress concentration factor for different types of welded joints after [9.1]

9.2 Stress concentration factor in welding cords The stress concentration factor is generally and in particular for a welded joint, defined by the ratio between the local maximum stress and the gross stress. This stress concentration factor depends not only on the radius at the weld toe U and 187

188

G PLUVINAGE

the connecting angle \ but also on the geometry of the welded joint and the loading mode. Different formulae for the stress concentration factor are proposed within the literature. Some simplified formulae do not take into account the entire range of parameters.

Endurance limit VD (MPa) 160 140 120 <

100 80 100

110

120

130

140

Connecting angle < (degree) Figure 9.2: Influence of the connecting angle on the fatigue limit at 2*106 cycles, after [9.2].

9.2.1 INFLUENCE OF THE CONNECTING ANGLE Richards [9.2] has shown that the fatigue limit increases linearly according to the connecting angle of the cord on a butt welded structure, see figure 9.2. 9.2.2 INFLUENCE OF THE RADIUS AT THE WELD TOE From a study of cruciform welded joints, using a finite-element method, Skorupa et al.[9.3] have developed an equation for evaluating the elastic stress concentration factor kt, taking into account the radius U at the weld toe and the thickness t of the plate. t 0.592 . (9.1) k t 0.217 U



Studies by Ikeda et al [9.4] show that as the radius at the notch tip decreases there is a corresponding decrease in fatigue strength, figure 9.3. These results have been

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

189

determined from tests carried out on cruciform joints (K2) with high yield strength (780 MPa). 9.2.3 INFLUENCE OF THE CONNECTING ANGLE \ AND THE RADIUS AT THE WELD TOE Studies carried out on machined samples with profiles similar to the welded joints have been achieved by varying the radius U and the connecting angle \ of the simulated joint. These studies have shown that an increase in the notch radius or a decrease in the connecting angle reduces the stress concentration effect considerably. The influence of U becomes negligible when T is smaller than 10° or 15° and the influence of \ becomes negligible either for a value higher than 40°, or when U exceeds 6~10 mm. Fatigue strength reduction factor kf

0.8 < = 15°

0.6

< = 30°

0.4

< = 45° < = 60°

U

0.2 0

1

2

3

4

Weld toe radius (mm) Figure 9.3: Influence of the radius at the weld toe on the decrease in fatigue strength after [9.4].

Niu and Glinka [9.5] assessed the stress concentration factor kt for a cruciform welded joint by means of the formula ( \ is the connecting angle in radians): 0.469 · § ¸ . 0.572 .¨ 0.217 t (9.2) k t 1  0.512.\ ¨ ¸ U © ¹



9.2.4





INFLUENCE OF THE CORD GEOMETRY

Nihei et al. [9.1] determined the stress concentration factor according to the geometric parameters of the cord height H, thickness t, U and \ (in radians) from the formula: kt = 1 + f (\) (D - 1),

(9.3)

190

G PLUVINAGE

with :

D

H º 1 ª «¬ U. 2.8 '1  2 »¼

1  exp 0.9\ . ' 2 1  exp 0.4S . 2

f \

'1 9.2.5

0.65

2 H  t t

and '1

,

,

2 H  t 2 H . t

INFLUENCE OF THE GEOMETRY AND LOADING

For the determination of the stress concentration factor, Machida [9.6] has taken into consideration the geometry and the type of loading. For three joint geometry’s, he proposed a formula for the stress concentration factor that may applied in the intervals of the U/t intervals (0,0025 d U/t d 0,2 and 0,26 d U/t d 5,2).

x cruciform welded joint subject to axial tensile loading ª § L ·º U 0.309 1 , k t ,1 1  f \ . exp «0.17  0.14.¨ ¸». © t ¹¼ t ¬



f (\ )



(9.4)



0.97  5.2 *10  4  0.005.LnL * , § 0.019 · ¸¸.P 1  ¨¨10.1  U ¹ ©

ª­ ½º 9 · § exp «® U .¨12.2  ¸  5.3  0.6 LnL * ¾.»\ . L*¹ © «¬¯ ¿ »¼ where L is the half - length between the axes.

with P

x T joint subjected to a bending (stress) kt2 ª § L ·º U 0.309 1 , k t ,2 1  f \ . exp «0.16  0.14.¨ ¸». © t ¹¼ t ¬



f (\ )





1.02  7.6 *10  4  0.00068.LnL * , § 0.027 · ¸¸.P 1  ¨¨ 9.8  U ¹ ©

(9.5)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

191

ª­ ½º 8.6 · § exp «® U .¨12.6  ¸  4.8  0.76 LnL * ¾.»\ L * © ¹ ¿ ¼» ¬«¯ r* = r/t and L* = L/t.

with P

x cruciform joint subject to a bending stress : kt





0.5. k t ,1  k t ,2 .

(9.6)

Stress concentration factor kt Finite element Machida

4

Glincka Skorupa Nihei

3 2

1 0 0

1

2

3 4 Weld toe radius (mm)

Figure 9.4 : Influence of the radius at the weld toe on the elastic stress concentration factor for a T joint.

9.2.6 COMPARISON OF THE ANALYTIC FORMULAS WITH A FE METHOD Finite-element calculations have been performed for T welded joints whose geometry is inset in figure 9.4, with a view to determining the influence of the weld toe radius, as well as the influence of the connecting angle. In figure 9.5, it is noted that the formulas of Nihei [9.1], Skorupa et al [9.3], Niu and Glinka [9.5] and Machida [9.6] not only give connecting angle but results also results related to the numerical calculations, as far as the influence of the radius at the weld toe is concerned. The major difference between the analytical formula and the finite-element calculations is that concerning the influence of the connecting radius. The influence of the stress concentration in an end to end joint compared with a plan joint is evident in figure 9.6.

192

G PLUVINAGE

Such a joint shows an increase in factor of 2.4 in the elastic stress concentration. We see that the fatigue strength of the planed joint is quite high compared with that of the rough joint. However the product of the gross stress and the stress concentration factor does not allow one to encounter the curve of the planed joint; the latter may actually be used as a reference supposed incorporating the effect of residual stresses. Element ˇFinite lˇ ments finis

kt

ˇMachida quation de Machida ˇGlincka quation de Glinka ˇNihei quation de Nihie

3

2

1 10

20

30

40

50

60

Weld toe radius (mm)

Figure 9.5: Influence of the connecting angle on the elastic stress concentration factor in a T joint.

9.3 Fatigue strength reduction factor The use of the stress concentration coefficient kt produces pessimistic fatigue predictions in relation to the experimental results. This is especially true for low radii at the notch tip. For this reason, we use another empirical coefficient in fatigue to characterise the notch effect. It is symbolised kf and denoted fatigue strength reduction factor. It is determined during similar tests and defined by: kf

VD V D, n

(9.7)

where VD is the fatigue limit on smooth specimens while VD,n is the fatigue limit on notched specimens. Some equations encountered in the literature allow one to determine the coefficient of stress concentration in fatigue, according to kt. A generalisation of the fatigue notch factor according to the number of cycles is now used:

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

k f N R

'V N R 'V n  N R

,

193

(9.8)

where 'V (NR)is the stress amplitude on unnotched specimens, 'Vn (NR) the stress amplitude on notched specimens. Stress range (MPa)

375

250

125

Erased welded joint Non erased

0 105

Local Stres range 5.105 106 Number of cycles for fatigue life duration

Figure 9.6: Influence of the stress concentration in an end to end joint compared with the plan joint.

The fatigue notch factor increases with increase in the number of cycles and tends asymptotically towards the value of kt for a large number of cycles, as can be seen in figure 9.7. The relationship between the elastic stress concentration factor and the fatigue notch factor kt may be classified in three categories depending on the assumption used: that is, models using empirical relations and based on the concept of mean stress for a certain distance, and models based on non- propagating stress for a short crack initiated from a notch.

9.4 Standard methods for the design against fatigue of welded components Although extensive work has already been conducted, prediction of the fatigue strength of welded structures is still a widely open subject. In practice design engineers tend to adopt simplified methods for evaluating fatigue life. These methods often have poor accuracy and there is a need for the use of computational methods which, although sophisticated, are more accurate and reliable.

194

G PLUVINAGE

These methods are generally based on the ‘Hot spot approach’ concept which considers that only the maximum stress plays the essential role. This derives directly used from the deterministic, general science approach ofthe19th century. The most widely used are: a) the linearised stress adopted methods code (BS Standard), b) the Eurocode III, c) the geometric and effective stress approach (proposal IIW). Recent methods considered any fatigue initiation and fatigue crack initiation is relevant to local approaches. 2.50

Fatigue strength reduction factor kf STEEL E 360 ; BUTT JOINTS

2.25

2.00

1.75

1.50 2.105

6.105

1.106

Number of cycles for fatigue life duration Figure 9.7: Variation of fatigue reduction factor versus number of cycles (butt joints).

9.4.1 LINEARISED STRESS METHOD This method is used in UK code BSI PD 6493 (1991) (9.7). This method is based on the determination of an effective stress range called the linearised stress range. This stress range is determined from a fatigue resistance reference curve where fatigue life duration is obtained along the abscissa. Generally, a structure is considered to be subject to superposition of the following loading conditions: ía primary stress owed to mechanical loading, ía secondary stress owed to thermal loading and residual stresses, ía peak stress induced by geometrical discontinuities, ía bending stress owed to misalignment.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

195

The stress distribution obtained in the thickness direction is linearised in a conservative manner as shown in figure 9.8. This linearisation procedure gives rise to a tensile stress range 'Vt, and a bending stress range 'Vb.

Figure 9.8: Linearised stress method showing the definition of the stress range 'V1 and 'V2.

The effective bending stress takes into account the misalignment stress. In this respect, a misalignment factor kdes is defined. The effective bending stress is given by 'V’b :

' V 'b

' V b  ' V t * k des .

(9.9)

Fatigue reference curves are different according to the different welds classes Q1 to Q10. These fatigue reference curves are presented using a double bi-logarithmic scale: curves are available for both steels and aluminium alloys. This method has no reference to material and R ratio and the influence of mean stress on fatigue reference is hidden by the presence of residual stresses which decrease during cyclic mechanical loading. 9.4.2 EUROCODE III [9.8] According to this code, the fatigue life duration of a welded structure depends on the stress range and of the category of the weld. A safe design satisfies the following relationship: 'V R

Jm

J s .' V a ,

(9.10)

where 'VR is the effective fatigue stress range (for random loading, this value is obtained using the ‘rainflow method’ and Miner’s rule, Js et Jm are safety factors.

196

G PLUVINAGE

Stress range (MPa) 1000

Q10 Q7 Q4

100

Q1

10 0

4

104

5

105

6

7

106

107

8

108

Number of cycles for fatigue life duration Figure 9.9: Fatigue reference curves according to British standard BS 6493 (only those relative to classes Q1, Q4, Q7 and Q 10 are presented).

The safety factor for the material Jm takes into account the scatter in fatigue resistance due to geometrical variations, stress concentrations, residual stresses and damage due to the weld process. The safety factor for loading Js takes into account the stochastic process of mechanical loading and conversion of load to stress. All welded structures are classified into detailed categories. These categories integrate process and control procedures, in particular taking into account pre existing cracks. These classes are expressed by a number Xcl which corresponds to the fatigue resistance for 5 million cycles and is expressed in MPa. For each class the fatigue reference curve is obtained by division into three parts. The curve is plotted as bi-logaritmic graph of stress range 'V versus number of cycles to failure. íPart 1: a line of slope (left and side) m=3 up to 5*106 ; íPart 2: a line of slope m=5 from point 5*106 ; where Xcl is the number associated with detail class; íPart 3: a horizontal line from 10 8 cycles. The curves shown in figure 9.10 have been established from numerous experimental results performed on specimen of a thickness of approximatively 15 mm. Eurocode can be used for welded joints with residual stresses and high R ratios but it does not take

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

197

into account the material and the stress ratio. For thickness values higher than 25 mm, the code leads to inappropriate safety factors.

Category number = Stress range for 2.105 cycles

Stress range (MPa)

1000

140 100

1

75 3 36 1 5

10 104

105

106

107

108

Number of cycles for fatigue life duration Figure 9.10: Fatigue reference curve according to Eurocode III (Only fatigue reference curves relative to detail classes 36, 71 and 140 are presented for clarity).

Stress range (MPa) 200

Best fit line for experiments

100

0

EUROCODE III 105

106

107

108

Number of cycles for fatigue life duration Figure 9.11: Wöhler curve for a T symmetric weld joint in steel E36, thickness 8mm, subject to bending, R = -0.1. Comparison of experimental results with Eurocode fatigue reference curve.

198

G PLUVINAGE

For welded elements with a thickness greater than 25 mm an empirical formula is used to correct the effective stress range: 25 0.25 , (9.11) 'V R 'V N B



where 'VN is the effective stress range given by the reference curve. Comparison of experimental results with the Eurocode fatigue reference curve for a T symmetric weld joint using steel E36 of thickness 8mm, subject to bending at R = -0.1 shows the conservative nature of the of Eurocode III approach, see Figure 9.11. 9.4.3 GEOMETRICAL AND NOTCH EFFECTIVE STRESS RANGE [9.9] The international welding institute has proposed an ISO standard in document XIII 1539-96/XV–345-96 untitled ‘Recommendation for fatigue design of welded joints‘ [9.9]. This design code is also based on a fatigue reference curve for each weld class being similar but slightly different to the Eurocode III standard. The difference can be found in the applied stress range used which can be either the geometrical stress range or the notch effective stress range. The geometrical stress range is related to the geometry and the type of weld joint. It takes into account the misalignment effects, residual stresses and scale effects. The geometrical stress is determined from linearisation of the true stress distribution obtained by a finite Element method or using strain gauges. The procedure consists of building a linear distribution from the stress range value of the particular points d1 and d2 where d1 and d2 are chosen according to the plate thickness B, d1=0,4 B and d2-d1= 0,6 B. The stress amplitude at point d1 and d2 is measured with very small strain gauges, the grid length being smaller than 0.2 B. Generally the theoretical stress distribution is linear for tension and quadratic for bending. Stress

Maximum stress Geometrical stress Hot spot

F

Strain gauges d1 d2 Figure 9.12: Definition of the geometrical stress in a weld joint.

F

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

199

Extrapolated values at the weld toe give the geometrical stress precisely. In the case of a geometrical discontinuity leading to a stress concentration, the effective notch stress is obtained from the real stress distribution computed by finite element method with the following assumptions: linear behaviour and welded toe radius equal to 1mm. Fatigue reference curves are linear in a bilogarithmic graph with a slope equal to 3. This line passes through the point [2*10 6 ; Xcl] where Xcl is the weld class number (in MPa). Detail categories are more conservative than for the Eurocode III approach. The method takes into account the R ratio if R< 0.5 by modification of the detail class number. This is obtained by multiplying with a factor which is dependent upon R ratio according to the following table9: R

A

B

C

R <-1

1,6

1,3

1

-1< R >0.5

0.4R+1,2

0.4+0.5

1

R > 0,5

1

1

1

Table 9.1 : Values of the factor used to change the weld class to take into account the R ratio.

A) base material and welded joints without residual stresses and secondary effects; B) simple welded joints, thin plates and simple joints; C) complexes structures and thick plates. 9.5 Innovative methods for the design against fatigue of welded joints

9.5.1 LOCAL STRAIN METHOD [9.10] It has been observed that due to plastic relaxation and subsequent redistribution of the stress field in the region near the stress concentration, the strain field close to the weld toe is a suitable parameter which may be used to evaluate the fatigue strength of welded joints. This is the basic principle of the local strain method. The technique takes into account all the factors that can influence the strain, e.g., ístress concentration due to the weld toe radius; ílocal plasticity; ímisalignments of the structure which cause secondary bending. This method offers an easy and rapid experimental method to predict fatigue life duration of welded joints. A fatigue reference curve in terms of the applied strain range versus the number of loading cycles is required. The principle of this method is based on the following assumptions: íthe radius of the weld toe is zero,

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íthe behaviour of the material is elastic remote from the weld toe, the stress distribution is given by the William’s solution [9.13]. The first assumption is highly conservative because the radius at the weld toe is always higher than this value and consequently the stress concentration is less. However due to the difficulties in measuring this radius, this assumption is helpful. The second assumption neglects the plastic relaxation at the notch tip of the weld toe and is also conservative. However if the notch plastic zone is small enough, this assumption can is satisfactory as confirmed by finite element, analytical methods and by the moire-holographic method. log (Hyy) Strain distribution according to William’s solution 2;5 mm

log r Strain gauges

F

F Figure 9.13: Principe of the local strain method.

According to the third assumption above, stress and strain are given by the following relationship : K *H , (9.12) H ij D  2Sr where KH is the notch strain intensity factor, D the exponent of the stress or strain singularity which according to Williams, depends on the notch angle (D = – 0.316 for an angle of < = 135 °). A unique measurement of the strain in the elastic region is enough to characterise the stress state and fatigue resistance of the structure. It can be evaluated by means of some (4-5) electrical strain gauges with a 3 mm grid length bonded along the axis at about 2.5 mm from the actual weld toe and loading the joint with nominal fatigue loads (maximum and minimum stresses) and comparing the difference of measured strains with the reference curve. The strain range is measured when the material is cyclically stabilised and given by: H max  H min , (9.13) Ha 2

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

201

where Hmax and Hmin are respectively the value of maximum and minimum strain. Strain range (PH) 10000 T symetric Welded joint

HLE STEEL Thickness B= 8 mm E 36

1000

100 104

105

106

107

Number of cycles for fatigue life duration Figure 9.14: Fatigue reference curve for steels E 36 and HLE (symmetric T Joint B = 8mm).

The fatigue reference curve is determined for one type of joint. This fatigue reference curve has a universal character, it can be used for any kind of welded joint. Influence of geometry, thickness and weld joint type are incorporated into the notch strain intensity factor which characterises the strain distribution at the notch tip (for a given weld joint angle). Strain range (PH) 10000 25 mm 5mm 1000

100 105

106

107

Number of cycles for fatigue life duration Figure 9.15: Influence of thickness on the local strain, cruciform welded joints construction steel.

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An example of such a reference curve is given in figure 9.14 for steels E 36 and HLE (symmetric T Joint, B = 8mm). This curve exhibits a reference value of about 600 PH for 2.*106 cycles. The fatigue reference curve is not very sensitive to weld joint type and thickness as can be seen in figure 9.15 and 9.16. However, the fatigue reference curve is sensitive to material and load ratio R. For this parameter a linear dependence relationship as given in figure 9.17 has been found. Strain range (PH) 10000

1000

100

104

105

106

107

Number of cycles for fatigue life duration Figure 9.16 : Influence of type of joints on the local strain, cruciform and angular welded joints, identical steel.

9. 5.2 THE VOLUMETRIC METHOD [9.11] The volumetric approach proposes that the fatigue process requires a given volume. Note the hot spot approach reduces this volume to one point. In this volume, the effective stress is some kind of average stress. The limit of the fatigue volume is determined by the distance of the minimum relative stress gradient. This definition has been successfully applied to different kinds of material and loading modes. In order to take into account the type of loading and scale effect, the stress distribution is "weighted" as a function of the distance, stress gradient, and triaxiality. This is one of the major benefits of this method which allows a unique fatigue reference curve for any cases of loading (tension, bending or torsion). The effective stress is obtained from the average value over this effective distance of the ‘weighted’stress distribution. The ‘weighted’ stress is a combined value of the stress, the distance to the weld toe and the relative stress gradient. The stress distribution at a symmetric T weld joint has been computed by finite element using the cyclic stress-strain curve and is presented in a bi-logarithmic graph as shown in figure 9.18. From this figure 4 zones can be defined:

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

203

ízone I : near the weld toe where the stress increases up to its maximum; ízone II : transition zone where stress decreases after the maximum; ízone III : a zone where the notch stress intensity factor governs the stress distribution; ízone IV : far from the weld toe where the stress plays no major role in the fatigue phenomenon.

Strain range at 2.105 cycles 600

500

400

300

200 -1.2

-1.0

-0.8

-0.6 .

-0.4

-0.2

0 Stress ratio R

Figure 9.17: Influence of R ratio on the strain range corresponding to 2.10 2 cycles.

The fatigue phenomenon is influenced by the loading mode, specimen thickness and weld geometry, any of which can be attributed to the role of the stress gradient. For this reason the stress distribution is modified taking into account the value of the relative stress gradient as defined by 1 dV F . , (9.14) V x dx where F is expressed in mm-1 and x distance from weld toe. It is generally assumed that the plastic zone at the notch root is cylindrical. This assumption needs to be verified using a finite element method for the weld toe. By analogy the fatigue process volume is also cylindrical and the diameter of this cylinder is precisely the effective distance Xef.

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The effective stress determination is based on the fact that the fatigue process volume is a highly stressed region. The limit of this highly stressed region has been found by trial and error and is associated with the minimum relative stress gradient. This limit has been verified by numerous experimental results.

Figure 9.18: Stress distribution at weld toe of a symmetric T joint calculated by finite element method, bil-ogarithmic graph.

The influence on any stress value inside the fatigue process volume is a combination of the value of the local stress value, the distance to the notch tip where initiation takes places. In addition, the stress value is dependent upon the relative stress gradient which to take into account geometrical, scale and loading effects.All these combined influences can be represented by a weighted stress defined as follows ' V *ij

' V yy I x, F ,

(9.15)

where I(x,F) is the weight function. This function can also be written as :

I x, F 1  r.F ,

(9.16)

where r is distance from weld toe tip.The effective stress is the average value of the stress distribution inside the fatigue process volume X ef 1 (9.17) ' V ef ³ ' V *ij.dx , X ef 0 which can be written as follows : X ef 1 ' V ef (9.18) ³ ' V yy.1  rF dx , X ef 0

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

205

where 'Vyy is the normal opening stress range.

400 350

T joints, nominal stress smooth butt joints T joints, effective stress

Reference curve

300 250

Method results

200 150 100 50 0 100000

Experimental curve

1000000

N

10000000

Figure 9.19: Application of the volumetric method for the prediction of fatigue life of a symmetric T welded joint, ( HLE steel, thickness 8 mm). Stress amplitude in MPa.

An experimental verification of the application of the volumetric method for the prediction of the fatigue life duration of a symmetric T welded joint in HLE steel, thickness 8 mm, can be seen in figure 9.19. Predictions given by this method are relatively good and require only the fatigue reference curve and cyclic stress-strain behaviour of the material to be introduced in the finite element method. For convenience it is often assumed that the notch radius at the weld toe is zero. Atzori et al [9.10], Lazzarin et al [9.12] and Verreman et al [9.14] have used this assumption. This approach is justified by the fact that generally the radius at the weld toe is very small compared to other geometrical parameters and its value is subject to large scatter. Consequently, we assume that the stress distribution at the weld toe is a singularity. An example of such distribution can be given for a cruciform welded joint, see figure 9.20. 9.5.2.1 Weld defect with an infinite acuity In the case of elastic behaviour the stress field for a 2-dimensional case results from the superposition of symmetric and anti symmetric stress fields.

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< = 135 ° T= 0° T = 22.5°

B

F

F

h L

Figure 9.20: Geometrical parameters for a cruciform welded joint.

For the case of cruciform welded joint, the stress field can be described by Williams‘s solution [9.13] and is governed by the notch stress intensity factors: sin O1.qS  O1sin qS 0 ,

sin O 2 .qS  O 2 sin qS 0 .

(9.19)

In polar co-ordinates, (r,T) the symmetric stress field is singular and given by :

V TT V rr V rT

P

r O1  1. K *I .P , 2S 1  O1  F 1 1  O1 1

.

1  O1 . cos1  O1 T  F 11  O1 . cos1  O1 T 3  O1 . cos1  O1 T  F 11  O1 . cos1  O1 T 1  O1 . sin 1  O1 T  F 11  O1 . sin 1  O1 T

.

(9.20)

where q is related to notch angle < by the following relationship, see figure 9.21.

\



S 1  2q 

The anti symmetric stress field is also singular and also given by :

V TT V rr V rT

r O 2  1. K *II .P , 2S 1  O 2  F 2 1  O 2 1

.

(9.21)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

207

 1  O 2 . sin 1  O 2 T  F 2 1  O 2 . sin 1  O 2 T

P

 3  O 2 . sin 1  O 2 T  F 2 1  O 2 . sin 1  O 2 T 1  O 2 . sin 1  O 2 T  F 2 1  O 2 . sin 1  O 2 T

. (9.22)

y

< x qp/2 T

r

VTT VrT Vrr

Figure 9.21 : Co-ordinate system for the weld toe in this case the radial stress is considered along the direction T = 112..5°.

The radial stress is given using superposition principle from notch stress intensity factors

V rr

K *I .0.423 r 0.326  K *II .0.423 r 0.302 ;

(9.23)

The fatigue resistance curve is determined using the amplitude of the notch stress intensity factor in mode I, ¨K*I versus the number of cycles. This approach changes only the presentation of curve and presents the results in units of MPa m (1-O). 9.5.2.2 Weld defect with a finite notch radius

In the case of a finite notch radius the stress distribution is not singular and exhibits a maximum. In order to represents this maximum, the singular distribution is kept but the origin is shifted to a distance equal to r0 in order to find the maximum stress for r = 0 . This method has been used by Creager and Paris [9.15], Glincka [9.16] for a 1/¥r dependence and by Lazarrin and al [9.12] for a r-D dependence. In the last case, the stress distribution is governed by the notch stress intensity factor range for an infinite acuity ¨K*.

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9.5.2.3 Lazarrin et al solution [9.12]

Lazarin et al have used this method for prediction of life duration of cruciform weld joint (see figure 9.20). They modified the Williams’s solution in the following manner :

V TT V rr V rT U z 0

A

V TT V rr V rT U

3  O1  F 11  O1 , 1  O1  F 11  O1

. 0



r P 2 1 K *II . A.B 2S r1  O 2 r 0 0 1

.





cos 1  P1 T B





 cos 1  P1 T





.

(9.24)

sin 1  P1 T

In this formula, r0 is a distance related to the notch root radius. 9.5.2.4.Verreman’s solution [9.14]

The stress normal to the notch plane divided by the gross stress is equal to:

V yy V yy

O0



a D , B

(9.25)

where O0 is a stress distribution parameter, a notch depth B the specimen thickness and Dsingularity exponent. The stress distribution is given by : V yy a i a D i 4 . (9.26) ¦ Oi B B V yy i 0





If the term a/B is small equation (9.26) reduces to equation (9.25) The stress magnification MK is given by : a D . M k O 0.l 0 B



(9.27)

Verreman has made the assumption that the stress normal to the notch reaches the value of the gross stress at a distance proportional to the thickness to the power D where D is the singularity exponent. The notch stress intensity factor is then given by: K *I

V g H 0 BD 2S .

(9.28)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

209

The fatigue reference curve can also represented by the notch stress intensity factor range versus the number of cycles to failure, the units of this parameter are also MPamD. Log (Vyy/Vg) 6

Y

4 D

2

1 E0.BD.—(2S) 1 log (r/B) 10-5

10-4

10-3

10-2

10-1

Figure 9.22: Example of the non dimensional stress distribution versus the non -dimensional distance. The pseudo- singularity region is governed by the notch stress intensity factor.

9.6. Application of the effective stress concept to fatigue corrosion of welded joints Corrosion in combination with fatigue loading gives rise to a synergy which affects both fatigue crack initiation and propagation [9.17]. At the weld toe specific phenomena such as blunting, dissolution,and hydrogen embrittlement may occur. Studies concerning the corrosion fatigue of welded joints consider the corrosive environment as a constant (independent) factor that characterises testing conditions only. Such approaches do not satisfactorily take into account the role of the environment and the use of a simple reduction factor for fatigue life duration is not appropriate. Specific studies involving individual welds are required. 9.6.1 EXPERIMENTAL APPARATUS FOR CORROSION FATIGUE TESTING OF WELDED JOINTS Welded steel T joints with an asymmetric toe were chosen for the corrosion fatigue studies. These joints were manufactured according to requirements of Eurocode III with E 36 steel. Specimens have been made from steel sheets, width 45 mm, thickness 6~8 mm with a geometry given in figure 9.23. The specimens were tested under three point bending conditions. The scheme of corrosion fatigue test rig is shown in Figure 9.24.

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180 mm

8mm 10 mm 44 mm 6 mm 180 mm

6 mm Figure 9.23 : Geometry of the non- symmetric T welded joint.

The tests were conducted under constant level of cyclic stress and a stress ratio R = 0.1. The frequency of loading was 2 Hertz. The processes of fatigue crack nucleation and growth were observed using an optical microscope under stroboscopic lighting. F 8

4 6 4

5

3 F/2

7

2 F/1

Figure 9.24: Schematic of corrosion fatigue tests Legend for figure 9.24 : 1 íspecimen; 2 í corrosion cell; 3 í corrosive environment; 4 í grips; 5 í reference electrode; 6 íauxiliary electrode; 7 í potentiostat; 8 íXY-recorder

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

211

During tests were held constant electrochemical conditions using a potentiostat, reference and auxiliary electrode arrangement. Periodically the solution was refreshed to maintain constant pH conditions. The general features of corrosion fatigue process were as follows. During the initial stage of the tests for both E36 and HLE joints, splitting of the welded joint under cyclic tension is observed. As a result of this a crevice is formed, located at the weld metal/parent plate interface. This process occurs within the first few percent total lifetime. The crevice may be considered as a special type of stress concentrator from which fatigue crack nucleation occurs. During the second stage fatigue crack nucleation and growth in the base material occurs consuming the majority of life. The duration of this stage consists of 60~80% of the total life time depending on the level of cyclic stress and the environmental conditions. Fatigue cracks nucleate near the weld interface in the heat affected zone of the parent material in a direction of at some angle with respect to the plane of maximum tensile stress. Despite some scatter this angle can be accepted as constant (T = 17°) for all testing conditions applied. 9.6.2 EXPERIMENTAL RESULTS It was observed that the fatigue crack growth stage is relatively short and characterised by increase of fatigue crack growth rate, da/dN, with increase in crack length a, see Figure 9.25.

Stress range (MPa) air corrosion

300

200

100 104

105 106 Number of cycles to fatigue life duration

Figure 9.25 : Wöhler curves for E36 steel in air and in a corrosive environment, (3,5% NaCl, 20°C, frequency 5 Hz).

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These observations suggest that the lifetime of the given type of joint is determined by the time for fatigue crack nucleation and growth to a length of about 1mm. It may be seen in Figure 9.26 where the corresponding endurance curves versus lifetime up to a =1mm. are presented. The influence of corrosion potential of polarisation on fatigue crack initiation depends upon the nature of material but generally decreases with the increasing of absolute value of polarisation potential, see figure 9.27.

9.6.3 MODELISATION OF SYNERGISTIC EFFECT BETWEEN CORROSION AND MECHANICAL LOADING The following equation may be used to related corrosion and loading to fatigue lifetime; º m ª M § 1 ·N* .¨ ¸. ³ iN dN » C 0 cst , (9.29) « 'V ef z. f .U © Z ¹ 0 ¬« ¼»





where Vef is the effective stress range, m and C0 are constants which depend of the material- environment system, M molecular mass, z number of electrons which participate in metal dissolution, Z frequency, i current density, N* number of cycles to initiation (corresponding to crack length a*).

Stress range (MPa) 160

air corrosion

140 120 80 104 105 106 107 Number of cycles for fatigue life duration Figure 9.26: Wöhler curves for HLE steel in air and in a corrosive environment.

For a constant current density i cor = const , equation (9.29) can be written : m ª M §1· º .¨ ¸. I cor N *» C 0 cst . (9.30) 'V ef « U Z z . f . © ¹ ¬ ¼





FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

213

For given values of icor and Veff the number of cycles to initiation is given by: N*

§ · ¨ ¸ z. f .U C0 ¸ . .Z.¨ m¸ M I cor ¨ ¨ 'V ef ¸ © ¹





(9.31)

Good correlation between experimental and computing values for fatigue initiation has been noted for this analysis and test data. Number of cycles for fatigue initiation 106 E36 HLE

105

104

-900

-800

-700 -600 -500 Polarisation potential (mV)

Figure 9.27: Influence of Potential of polarisation on fatigue crack initiation ( Steel E36 and HLE).

9.7 Conclusion The elastic stress concentration factor depend on the weld toe radius, weld angle, loading mode and geometry. Specific analytical formulae from Nihei et al. [9.1], Skorupa et al.[9.3], Niu and Glinka [9.5] describe the influence of theses parameters on the kt value. The use of this parameter is limited by plastic relaxation and limitations of the ‘hot spot’ approach. Traditional design of welded joints against fatigue is made according to codes such as Eurocode III [9.8], the geometric and effective stress approach [9.7] and the linearised stress code [9.9]. The design principle is a specific fatigue reference curve for each class of weld excluding the influence of material properties. Innovative methods such as the local strain [9.10] and volumetric methods [9.11] take into account, in a precise manner, the cyclic material behaviour. Recent methods which consider fatigue initiation and crack growth is relevant to the local approach adopting the concept of notch stress intensity factor [9.13; 9.17]. The synergic effect between mechanical loading and corrosion in welded joints can be seen by applying a model which couples the effective stress and the corrosion current [9.17].

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Number of cycles for fatigue initiation (experimental) STEEL HLE a* = 0.4 mm Frequency 5 Hz ; Ph = 7 E = -980 mV

105

0 0

105

2.105

Number of cycles for fatigue initiation (predicted)

Figure 9.28: Comparison between experimental and calculated values for fatigue corrosion initiation (Steel HLE, E =-880mV vs SCE, crack length for initiation a*= 0.4mm).

REFERENCES 9.1 Nihei. M., Sasaki. E., Kanad. M. and Inagaki. M. (1981).‘Statistical Analysis on Fatigue Strength of ArcWelded Joints Using Covered Electrodes Under Various Welding Conditions With Particular Attention to toe Shape’. Transactions of National Research Institute for Metals, 23, N°.1. 9.2 Richards. K.G. (1969).’Fatigue Strength of Welded Structures’. The Welding Institute. 9.3 Skorupa. M., Braam. H. and Prij. J. (1987).‘Applicability of approximate KI solutions towards cracks at weld toes’. Eng. fract. Mech. 26, pp. 669~681. 9.4 Ikeda. K., Denon. S., Godai T. and Ogawa.T. (1978).’Improvement of the Fatigue Strength of Fillet Welded Joints in 780 N/mm2 High Strength Steel’. Welding Research International, 8, N°.1. 9.5 Niu. X. and Glinka. G., (1987).‘The weld profile effect on stress intensity factor in weldments’. Int. J. Fracture, 35, pp. 3~20. 9.6 Machida.R. (1991).SR202 Committee, Research on Fatigue Design and Quality of Welded Parts in Offshore Structures Shipbld. Research Assoc. Japan. N°. 395. 9.7 BSI Standard. (1991).‘Guidance on methods for assessing the acceptability of flaws in fusion welded structures’. PD 6493. 9.8 Eurocode 3. (1992).‘Design of steel structures’.National Application Document - Part 1-1: General rules and rules for buildings - Chapter 9: Fatigue. 9.9 ISO Standard Proposal. (1996).’Recommendations for fatigue design of welded joins and components’. IIW document XIII-1539-96/XV-pp 845~96. 9.10 Atzori.B., Blasi.G. and Pappaletttere.C. (1985).’Evaluation of fatigue strength of welded structures by local strain measurements’. Experimental Mechanics ,N°25-2, pp 129~139. 9.11 Boukharouba. T. Gilgert .J. and Pluvinage. G ; (1998). ‘ Role des Concentrations de Contrainte dans la fatigue des joints soudés’. Congrès AFIAP, Paris. 9.12 Tovo .R. Lazzarin.P. (1999).‘Relationships between local and structural stress in the evaluation of the weld toe stress distribution’. International Journal of Fatigue ,vol 21, pp 1063~1078. 9.13 Williams. M.L. (1952). ‘Stress singularities resulting from various boundary conditions in angular corners of plates in extension’. Journal of Applied Mechanics ,19 , pp 526~528. 9.14 Verreman. Y.E. et Nie.B. (1996).’Early development of fatigue cracking at manual fillet welds”. Fatigue and Fracture of Engineering Materials and Structures ,Vol N°19, pp 669~681. 9.15 Creager. M. and Paris P.C. (1967).’Elastic field equations for blunt cracks with reference to stress corrosion cracking’. International Journal of Fracture ,3 , pp 247-252. 9.16 Glincka .G. and Newport. A. (1967). ‘Universal features of elastic notch tip stress fields’. International Journal of Fatigue ,9, pp 143-150. 9.17 Panasyuk.V.V, Dmyktrakh.M.I., Pluvinage.G and Quylafku.G. (1999).’On corrosion fatigue emanating from notches : Stress field and electrochemistry’ (Conference on corrosion), Lviv.

CHAPTER 10 SHORT FATIGUE GRACK GROWTH EMANATING FROM NOTCHES ______________________________________________________________________ 10.1 Short cracks emanating from smooth surface For long cracks, the crack propagation rate is described by the Paris Law [10.1] which assumes that the fatigue crack growth is governed by the stress field at the crack tip. For numerous service cases, the stress range level is low and this stress field can be characterised by the elastic stress intensity factor, K. The Paris law is expressed by the following relationship: da (10.1) C.'K m , dN where a the crack length and N the number of cycles, C and m are constants. This fatigue propagation law concerns only cracks greater than about 1mm. Short crack growth rates not fall into this so-called Paris regime and differs due to the following points: í the crack propagation rate is discontinuous, firstly decreasing to a minimum value after crack growth accelerates until continuous Paris type of behaviour is observed; í for stress range values lower than the endurance limit, cracks can stop growing within several microns of propagation; í cracks do not continue to propagate at a stress range level lower that the endurance limit. Crack growth

.

Short crack regime

'V1 'V2

Long crack regime

'V1>'VD>'V2

Crack length Figure 10.1: Schematic showing the short and long fatigue crack growth regimes.

215

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Figure 10.1 illustrates the socalled short crack regime which characterise theses differences with the long crack regime. A distinction between short and long cracks is generally not made by consideration of their length but by the physical mechanism of propagation or non-propagation. Using propagation law description, we can distinguish: ílong cracks which obey the Paris regime, ímicrostructural short cracks; for crack lengths which follows a decreasing crack growth rate, íphysically short cracks which follow an increasing crack growth rate but different from the Paris Law. According to Miller et al [10.2] microstructural short cracks have a reduced crack growth rate or can be arrested by microstructural barriers such as inclusions or grain boundaries. The size of these microstructural short cracks is of order of several grains size and called microstructural barriers. Stress range (log'V)

'V= 'VD

PROPAGATION 'V

'K th

Sa . F V

NON-PROPAGATION

Microstructural short cracks

Physically short cracks

Long cracks

Microstructural barriers Figure 10.2: Description of the three types of cracks: microstructural short cracks, physically short cracks and long crack.

The crack propagation rate for this kind of short crack is given by the following equation [10.3]: da A.' J D .a b  a , (10.2) dN

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

217

A and D are material constants, 'J the shear stress range and ab the size of the dominant microstructural barrier. For physically short cracks, the crack growth rate increases from a minimum value to a value given by the Paris regime according to: da B.' J E .a  D , (10.3) dN B and E are material constants and D a threshold value. The distinction of these three types of cracks can also be made using the stress range for non-propagation. For this, a diagram similar to the Kitagawa diagram [10.4] is used (Figure 10.2), the difference being the incorporation of a microstructural short crack regime. For long cracks, the stress range 'V np for non-propagation is derived from the fatigue threshold 'Kth : ' K th ' V np , (10.4) Sa F V where FV is the geometrical correction factor . For microstructural and physically short cracks, the stress range for non propagation 'Vnp is always lower than the stress range for non-initiation which coincides with endurance limit VD. 'Vnp< 'VD. (10.5) Effective stress range for initiation

Stress range corresponding to fatigue threshold

SHARP NOTCHES

BLUNT NOTCHES Ucr

Notch radius

Figure 10.3: Influence of the notch radius on the effective threshold stress range: differences between sharp and blunt notches.

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10.2 Short cracks emanating from notches

Crack initiation from notches varies with notch radius and this leads notches being divided into two categories, i.e., sharp and blunt notches. Sharp notches with a very small notch radius can be considered as a crack and the initiation stress range is related to the fatigue threshold. When the notch radius increases, the notch plastic zone induces greater residual stresses at the notch tip and contributes to an increase in the effective fatigue threshold. Blunt notches at low stress levels exhibit an effective elastic stress range at the notch tip. The stress range for initiation can be considered as the endurance limit corrected by the stress gradient. The limit of the two regimes corresponds to the critical notch radius Ucr. Propagation of short cracks emanating from a notch differs also with the notch radius as shown in figure 10.4. With sharp notches, cracks initiate very rapidly at a level which corresponds to the true fatigue threshold, i.e., without any crack closure effect. This is due to the fact that there is no plastic zone wake behind the notch which otherwise induces crack closure. Crack propagation rate (da/dN) Short crack emanating from sharp notch Short crack emanating from blunt notch

Short crack emanating from smooth surface

Long crack regime a* Logarithm crack length (log a) Figure 10.4: Schematic showing short cracks emanating from sharp and blunted notches.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

219

Some authors have mentioned that crack growth rate can be higher along a short propagation length than the corresponding long crack growth rate because the effective stress intensity factor is higher in absence of crack closure effects [10.5]. For blunt notches it is common to observe a decrease in crack growth rate until a minimum which is generally higher than the corresponding minimum for a smooth specimen. For high applied stress eventually the crack can be arrested. An example of such behaviour of short crack growth is given for experiments using Single Edge Notch Tensile (SENT) specimens made in (French standard) steel E 26. This material has a chemical composition as shown in table10.1, the mechanical properties are described in table 10.2. Young’s modulus (MPa) 20600

Yield stress (MPa) 265

Ultimate stress (MPa) 410

Elongation % 30

Table 10.2 : Mechanical properties of(French standard) steel E 26.

%C

% Mn

% Si

%P

%S

%N

.20

0.42

0.26

0.05

0.05

0.08

Table 10.1: Chemical composition of (French standard) steel E 26.

Crack growth rate (mm/cycle) 105

Notch radius U = 0.06 mm

da/dN = B.('J)E.(a-D)

104 5 10 Crack length (mm) Figure 10.5: Crack growth rate versus crack length for crack emanating from a notch with a notch radius less than a critical value. 1

In these experiments fatigue cracks were initiate at net stress equal to 72 MPa for specimens exhibiting four different notch radii (0.06 ; 0.08 ; 0.12 and 0.14 mm). It was noticed that crack growth exhibited a logarithmic dependence with the number of cycles for notch radii less than 0.10 mm (figure 10.5). For notch radii higher than a critical notch radius Ucr, thecrack growth presents an initial non linear behaviour (in a bi-logarithmic graph), figure 10.6. Values of this critical

220

G PLUVINAGE

notch radius have been found by several authors, see table 10.3. Furthermore, it has been noted that this value is sensitive to the material tested. Authors Taylor et al [10.5] Jack and al [10.6 ] Frost [10.7] Swanson et al [10.8] El Bari et al [10.9]

Material Steel Mild steel Mild steel Aluminium Alloy Mild steel

Ucr (mm) 0.003 –0.04 0.25 0.06 O.15 0.10

Table 10.3: Critical notch radius values found by several authors.

Crack growth rate (mm/cycles) 104 Notch radius U = 0.12mm

105

Crack length a (mm) 0.5

0.1

Figure 10.6: Crack growth rate versus crack length for crack emanating from a notch with a notch radius greater than a critical value.

10.3 The role of the cyclic notch plastic zone

10.3.1 CYCLIC NOTCH PLASTIC ZONE AND CRITICAL NOTCH RADIUS The notch plastic zone can be computed using Finite Element Method and applying the cyclic stress-strain curve described by the Ramberg-Osgood relationship: 'H

'V 'V 1 n'  , E H'



(10.6)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

221

where E is the Young’s modulus, H’ the cyclic plastic strength coefficient and n’ the cyclic strain hardening exponent. For the steel E26 the following material constants are use: E (MPa) H’ (MPa) n’ 206 000 752 0,21 Table 10.4: Material constants for the steel E26.

The cyclic plastic zone is defined as the limit of the stress value equal to twice the cyclic yield stress. The cyclic plastic zone R’p is included in the monotonic plastic zone and is induced by the reverse load. The size of the cyclic plastic zone is an increasing function of the net stress and decreases with notch radius exhibiting a power dependence: E , (10.7) R' p B. U where B and E are constant (E = 0.14). In figure 10.7 the cyclic plastic zone is divided by the notch radius and plotted versus U. From this figure we see that the notch plastic zone is equal to two times the notch radius for the value Ucr. Taylor et al [10.5] assumed that the critical notch radius corresponds to the value for which the notch stress intensity factor (given by Creager’s solution) is practically identical to the stress intensity factor associated with the short crack. From this, they obtained an approximate relationship:

U cr

0.38. R ' p .

(10.8)

The situation in which the cyclic plastic zone ahead of the notch is approximately equal in size to the width of the notch itself corresponds to a situation where the plastic zone is indistinguishable from the crack itself (according to the principle that the plastic zone plays the same role as the notch).

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4

Ratio cyclic plastic zone /notch radius

3

2 1 Ucr

0 0

0.1

0.2

0.3 Notch radius (mm)

0.4

Figure 10.7: Cyclic plastic zone divided by the notch radius plotted versus U.

10.3.2 CYCLIC NOTCH PLASTIC ZONE AND DISTANCE WHERE THE MINIMUM OF CRACK GROWTH OCCURS. As previously described, short cracks are characterised by growth rate which fluctuate, i.e., accelerate and decelerate, before showing an increasing continuous growth rate. Redrawing figure 10.6 by plotting crack growth rate versus the crack length divided by the size of the cyclic plastic zone, we note that a minimum occurs when crack growth is equal to the cyclic plastic zone, see figure 10.8. This phenomenon has some analogy with the phenomenon of crack growth delay after a fatigue overload. This phenomenon is characterised by a crack plastic zone running through an overload plastic zone [10.10]. The size of the cyclic plastic zone corresponds to the distance at which compressive residual stresses are present inducing crack closure by plasticity, thereby reducing the effective stress intensity factor and consequently crack propagation.

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

223

Crack growth rate (mm/cycles) 104

Notch radius U = 0.12mm

105

Crack length a (mm) 0.5

0.1

Figure 10.8: Crack growth rate versus the crack length divided by the size of the cyclic plastic zone.

Similarly, we have the plastic zone of the short crack running through the notch plastic zone which can be greater if the notch radius exceeds the critical value. The minimum corresponds to the distance where compressive residual stresses are present, i.e., the size of the cyclic notch plastic zone. Below the critical notch radius, the monotonic crack plastic zone is greater than the cyclic notch plastic zone, compressive residual stresses are suppressed by the crack monotonic plastic zone and no delay occurs.

10.4 Stress intensity factor for short cracks and crack propagation

Generally the stress intensity factor range for a short crack has the following relationship: 'K

'V Sl . F V ,

(10.9)

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where FV geometrical correction, l the length of the small crack at the tip of the notch. f a  l W 1.12  0.23a  l W  10.56 a  l W 3

2

(10.10)

4

 21.74 a  l W  30.42 a  l W .

Lukas et al (10.11) have proposed: 'K

'V Sl . F V

Q. k t , 1  4.5 l U

(10.11)

where Q is a shape factor equal to 1.12 and kt the elastic stress concentration factor. This formula is valid for c §c· . (10.12) ¨¨ U ¸¸  U © ¹ min



The value (c/U) min is material dependent. 10-4

Crack growth rate dc/dN (mm/cycle) Notch radius U = 0.06 mm

10-5 23

24

25

26 'K (MPa—m)

Figure 10.9: Crack growth rate versus stress intensity factor range for a short crack emanating from a notch with a radius below the critical value.

The following relationships have been proposed by Smith and Miller (10.12) For l ” d

'K

k t 'V Sl . F V ,

(10.13)

For l > d

'K

k t 'V S a  l . F V .

(10.14)

FRACTURE AND FATIGUE EMANATING FROM STRESS CONCENTRATORS

225

where a is the notch depth and d the distance over which the notch stress field has no effect. d

(10.15)

0.13 a.U .

El Haddad, Smith, and Topper (10.13) have proposed : 'K

E'H S l  l 0 ,

(10.16)

with l0

1

S

.

 'K th Re

2

,

(10.17)

where'k th is the fatigue threshold and Re the yield stress. In figure 10.9 and 10.10, we have plotted crack growth rate versus the stress intensity factor range for a short crack emanating from notches with a radius above and below the critical value. In this regime, the Paris law does not describe crack propagation and crack growth rate exhibits a minimum for a notch radius above a critical value. 10-4

Crack growth rate dc/dN (mm/cycle) Notch radius U = 0.1 mm

10-5

23

24

25

26

27

'K (MPa—m Figure 10.10: Crack growth rate versus stress intensity factor range for a short crack emanating from a notch with a radius above the critical value.

10.5 CONCLUSION

Short fatigue cracks emanating from notches by fatigue do not follow the classical crack growth law for long cracks. The crack growth is widely influenced by residual stresses in the notch cyclic plastic zone. This leads to a situation of a minimum crack growth where the notch radius is greater than that of a critical value. This minimum occurs for a crack extension equal to two times the cyclic plastic zone size.

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G PLUVINAGE

REFERENCES 10.1 Paris. P.C . (1964).’The fracture mechanics approach to fatigue’ .Proceedings of 10th Sagamore Army materials conference, Syracuse University Press. 10.2 Miller . K.J., Mohamed.H.J., Brown.M.W. and de Los Rios E.R. (1986).‘Barriers to short fatigue crack propagation at low stress amplitudes in banded ferrite-perlite structure’. Small fatigue cracks ( Editor R.O Ritchie and L langford), pp 639í656. 10.3 Miller K.J.(1987). ‘The behaviour of short fatigue cracks and their initiation’. Fatigue and Fracture of Engineering Materials and Structures. Vol 10, N°2, pp 93~113. 10.4 Kitagawa. H. and Takahashi.S. (1976).‘Applicability of fracture mechanics to very small cracks or the cracks in the early stage’. Proceedings International Conference on the Mechanical Behaviour of Materials”(ICM 2), ASM, pp 627~631. 10.5 Taylor.D. Staniaszek. I.A.N. and Knott. J.F. (1990).‘When is a crack not a crack ? Some data on the fatigue behaviour of cracks and sharp notches’. International Journal of Fatigue , Vol 12 , N° 5, pp 397~402. 10.6 Jack.A.R. and Price. A.T. (1970).Int Journal of Fracture Mechanics, 6, p 401. 10.7 Frost.N.E (1960). Journal of Mechanical Science Engineering, 2 p109. 10.8 Swanson. R.E. Thomson. A.W and Bernstein.I.M. (1986). Metallurgical transactions, Vol 17 A. 10.9 El Bari. H. Sahli.B. , Fassi- Fehri.O. Gilgert.J. and Pluvinage.G. (1997).‘Utilisation de la distribution réelle de contrainte en fond d’entaille pour le développement d’un nouveau critère d’amorçage de fissure de fatigue’.3 ème Congrès National de Mécanique , S.M.S.M Tétouan, ,Maroc 10.10 Robin.C. Louah . M.et Pluvinage. G. (1983). ‘Influence of an overload on the fatigue crack growth in steels’.Fatigue and Fracture in Engineering Material and structures, vol. 6,pp 1~13, 10.11 Lukas.L. Kung.L , Weiss.B. and Stickler.R. (1986).‘Non-damaging notches in Fatigue’. Fatigue and Fracture in Engineering Material and structures, Vol 9, N° 3, pp 195~204. 10.12 Smith.R.A and Miller.K.J; (1978).‘Prediction of fatigue regimes of notch components’. International Journal of Mechanical Sciences, Vol 20, p 201 10.13 El Haddad .M.H, Smith.J.N and Topper.T.H. Fracture Mechanics.ASTM STP 677, p 274.

227

List of Symbols a

notch length

b

ligament size

b

ligament size

b'

Basquin’s exponent

B

thickness

c

flow stress

C

Compliance

C1

constant

C2

constant

D

Shaft diameter

E

Young's modulus

Es

secant modulus

G

linear strain energy release rate

I

Inertia

J

Path Integral

JIc

fracture toughness Elastoplastic strain concentration factor

kH

kV

fatigue strength reduction strain energy density concentration factor Elastoplastic stress concentration factor

kt

elastic stress concentration factor

M

Path Integral

M

bending moment

n

hardening exponent

NE

Notch effect

Nr

number of cycles to failure

Pl

limit Load

P

load

q

Fatigue sensitivity index

kf KU,W*

228

List of symbols r

distance

Rb

distance from plastic zone limit

Re

Yield stress

Rm

ultimate strength

RN Ry

distance of the neutral axis plastic zone

Ti

surface traction

U

Work done for fracture

Uel

Elastic work

ui

displacement

Upl

Plastic work

W

width

W

width

W*

strain energy density

W*c

critical strain energy density

W*c,0

critical strain energy density

W*N

net section strain energy density

Xef

effective distance

xk

unit vector in k direction

V gc

critical gross stress

VN H0 V’f D E ' 'm 'p

net stres

'Vef

effective stress range

't Hef

total elongation

elastic reference strain fatigue resistance constant constant elongation bending elongation tensile elongation

effective strain

230

List of symbols

Hmax Hy

maximal strain

* *(n)

path gamma function

Kel

elastic eta factor

K Kpl

plastic eta factor

k

ratio

O

constant

T U Uc V* Vf Vg Vl Vmax

angle

< F D'

deformation at yield stress

eta factor

notch radius critical notch radius référence stress local resistance gross stress local stress maximum stress slip line opening angle Relative stress gradient exponent of pseudo singularity

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INDEX Elastic stress concentration factor, 3 Elastic stress distribution, 21 Elastoplastic strain concentration factor 4 Elastoplastic strain concentration factor 44 Elastoplastic stress concentration factor, 4,5 Elastoplastic stress concentration factor, 44 Energy criteria, 91 Erdogan and Sih’s criterion, 87 Eta coefficients, 93 Eta elastic, 93 Eta plastic for non symmetric tensile specimen, 99 Eta plastic for pure bending specimen,98 Eta plastic, 93 Eta plasticfor CT specimen, 100

‘3IJ’ criterion, 146 Apparent fracture toughness, 107 Barnby et al, 127,128 Barsom and Rolfe correlation, 150 Basquin’s law, 171, 172 Bhattacharya et al 33 Boukharouba et al, 162 Brand, 158 Bridgman’s formula,115 brittle propagation, 141 brittle to ductile transition, 84, 141 Buch, 158 Carpinteri, 79 Charpy energy, 141 Charpy Impact test, 135 Charpy U, 137 Charpy V, 137 Charpy, 135 CheníPan, 23 Coffin's law, 179 Crack, 3 CreageríParis, 75 critical gross strain, 113 critical strain energy density, 111 critical time, 145 cyclic notch plastic zone, 222 cylindrical notched bar in tension, 41 Czoboly et al, 92

Fatigue corrosion, 209 Fatigue sensitivity index, 14 Fatigue strength reduction factor, 12 Fatigue threshold, 218 Feddersen’s diagram, 2 Firao, 93 Flavenot and Skalli, 179 Geometrical correction factor, 82 Glinka-Newport, 23 Griffith and Owen, 116 Griffith criterion, 3 Gross and Mendelson, 70 Gross stress, 39 Hardrath, 10 Harris, 161 Hasebe, 71 Heywood, 161 Hill’s solution, 29,31 Homogeneous slip lines field, 26 Hot spot approach, 8

Dang Van, 175 ductile initiation, 142 Dugdale’s plasticzone, 129 dynamic critical stress intensity factor, 3 effective distance, 10, 12 effective fatigue threshold, 157 effective strain, 119 effective stress, 12 ,10 effective volume, 12 El Magd, 115 231

232

INDEX

Hydrostatic pressure, 173 Impact resistance, 141 Infinite sharp notch, 4 Influence of notch radius 109 Instrumented Charpy impact test, 144 Intrinsic fracture curve, 89 Irwin’s plastic zone, 129 Irwin, 67

Makhutov'method, 52 MansoníCofin‘s law, 179 Mesnager, 137 Microstructural barriers, 216 Microstructural short cracks, 216 Mixed mode, 87 MolskiíGlincka’s, 48 MorozovíPluvinage, 52

J integral, 91

Net stress, 39 Neuber‘s rule, 46, 47 Neuber, 10, 14, 23, 46 Niu et al, 77 Notch angle, 2 Notch ductility factor, 125 Notch effects, 1 Notch radius, 2 Notch root plastic zone, 129 Notch sensitivity, 85 Notch strain intensity factor, 120 Notch stress intensity factor, 63 Notched plate in bending, 44 Notched plate in tension, 44

Key hole, 137 Kitagawa diagram, 217 Knésl, 73 Koe’s method, 51 Kuhn and Hardraht, 157 Kujawski, 23 Kumar et al, 23 Limit load, 27 Limiting bending moment, 30 Lin and Pin Tong, 71 Lin et al formula, 37 Linear fracture mechanics, 3 Load at general yielding, 145 Load at maximum, 145 Load at the end of brittle propagation, 145 Load at the start of brittle propagation, 145 Local stress fracture criterion , 140 Local critical strain, 126 Local strain fracture criterion, 122 Local stress criterion, 83 Logarithmic slip lines field, 25 Long cracks, 216 Low cycle fatigue, 179 Ludwik’s law, 46, 95, 97, 109 Lukas and Klesnil, 161 M integral, 55, 82 Macclintoch, 114

Paris law, 216 Peterson, 10, 14 Peterson’s model, 157 Physically short cracks, 216 Plastic collapse, 112 Plastic stress concentration factor, 45 Position of the maximum stress, 138 Potential of Polarisation, 212 Probabilistic fracture criterion, 67 Pseudo singularity exponent, 77 Pseudo strain singularity,111 Randall and Merkle,125 Relative stress gradient, 83 Sailors and Corten correlation, 150 Sanz correlation, 149

INDEX Schnadt, 137, 138 Sharp and blunt notches, 218 Sheppard, 162 Shi’s constant, 92 Shockey et al, 128 Short cracks, 216 Simple notch, 2 Slip lines, 24 Stieler, 161 StowellíHardrathíOhman, 51 Strain energy density, 91 Strain energy density distribution, 108 Stress intensity factor, 67 Stress triaxiality, 117 Sumpter, 93 Switek, 161 Tetelman and MacEvilly, 35 Tiimoschenko, 23 Topper and El Haddad, 162 Transition temperature, 140 Turner, 91

233

Usami, 23, 120 Verreman, 79 Volumetric, method, 7, 9 Wallim correlation, 150 Wang and Zhao, 158 Weibull stress, 66 Williams, 69 Wöhler’s curve, 7 Xu et al, 40 Xu Kewein, 40, 134 Ye and Wang, 158 Yi- Sheng wu, 48

[ parameter, 46