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

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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: Vmax > VN > Vg. (1.1) 1

G PLUVINAGE

2

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 '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

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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).

222

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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.

226

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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.

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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

Vgc

critical gross stress

VN H0 V’f D E ' 'm 'p

net stres

tensile elongation

'Vef

effective stress range

't Hef

total elongation

elastic reference strain fatigue resistance constant constant elongation bending 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