Influence of seawater and residual stresses on fatigue crack growth in CMn steel weld joints

Theoretical and Applied Fracture Mechanics 16 (1991) 135-144 Elsevier

135

Influence of seawater and residual stresses on fatigue crack growth in C - M n steel weld joints L. Bertini Dipartimento di Costruzioni Meccaniche e Nucleari, Uni~'ersity of Pisa, Via Diotisalci, 2, 56126 Pisa, Italy

Results are presented on fatigue crack growth of weld joints made of C Mn structural steel plates in both air and seawater. Tests were conducted to identify the behavior at different locations of the joints as the frequency, R-ratio and electrochemical potential are varied. Residual stresses in specimens with welds are evaluated to analyse the fatigue crack growth behavior. Satisfactory predictions are obtained by accounting for residual stresses and crack closure.

1. Introduction Medium and high strength steels with yield stress above 350 MPa are being used more and more for building structures exposed to marine environment. Particular attention is thus being given to analyzing "tension leg" or modern "jacket" offshore platforms and pipelines [1,2]. Because of the likelihood of failure of these structures that could lead to economical loss as well as dangering human lives, stringent design requirements are needed to enhance reliability. As a supplement to the traditional SN-curve approach [3,4], the "damage tolerant" methodology could provide additional information for inspection and repair. Due to the complexity of the stress states arising from the load history and to the aggressive environment, assessment of the offshore structure integrity by application of fracture mechanics would involve several parameters that account for the loading, material property and environment. Prediction of the life expectancy of welded tubular joints would require a knowledge of how the welding process affects the fatigue lives of the structure under consideration. Reported in the work to follow are Corrosion Fatigue Crack Growth Rate (CFCGR) test results pertaining to different locations of weldments made from C - M n steel and marine environmental conditions. An empirical/numerical approach is used to evaluate the residual stress distribution in the weld joints. A finite element analysis is then employed to analyze C F C G R in a compact tension (CT) specimen with and without the influence of weldment.

2. Description of experiment Detailed description of the CFCGR tests can be found in [5,6]. Chemical composition and mechanical properties of the C - M n steel used in the present study are given in Tables 1 and 2. Because of its weldability, this material is particularly suited for modern offshore structures.

Table 1 Chemical composition of C - M n steel C

Mn

V

Nb

S

P

0.004

0.97

0.08

0.039

0.005

0.011

0167-8442/91/$03.50 ~ 1991

Elsevier Science Publishers B.V. All rights reserved

L. Bertini / Fatigue crack growth in C-Mn steel weld joints

136

Table 2 Mechanical properties of C-Mn steel Yield strength %s (MPa)

Ultimate strength o-u (MPa)

Elongation e (%)

490

510

27.8

2.1. Compact tension specimens and weldments

Weldments are made from 20 mm thick plates. The Gas Metal Arc Welding (GMAW) technique called for Inertarc 3341.2 electrodes whose composition is similar to that of the base material. A "one-half vee" geometry was chosen so as to obtain a rectilinear Heat Affected Zone (HAZ) normal to the plate surface. The Post Weld Heat Treatment (PWHT) involved maintaining a temperature of 575 ° C for 25 minutes while furnace cooling was applied to the welded plates according to Det Norske Veritas (DNV) recommendations [7] for large scale welded tabular joints. Standard Compact Tension (CT) specimens are then machined to a width W = 60 mm and a thickness B = 12 mm such that the crack lies in the plane which corresponds to the longitudinal/transverse direction of the material. Figs. l(a) and l(b) show, respectively, the CT specimens for the base material and weld joint. Tests are conducted for three different locations of the crack, namely in the base material, middle of the H A Z and in the fusion zone. Two semi-circular side grooves with one (1) mm radius are machined on the weld joint specimen sides to enhance the crack to grow straight ahead. If a = a / W , the stress intensity factor expression for the weld joint CT specimen is given by [6]: K=

e(2 + ~) BW0S(l_ •),5 [-0.022 + 20.0 a - 99.61 ,~: + 215.2 a 3 - 210.4 c~4 +73.3 a 5]

6O

I ,

i

i

i

I

I

LJ

i

(a)

il

~4

(b) t

RESIDUAL STRESS J-CUT Z - - - -MEASUREMENT LSTRAIN GAUGE--' Fig. 1. Compact tension specimens for (a) base material and (b) welded joint.

(1)

L. Bertini / Fatigue crack growth in C-Mn steel weld joints

137

Table 3 Seawater environment Type

f (Hz)

R (O'min/o-m~ x)

Potential ( m Y Ag/AgCI)

a b c d

1.0 0.3 0.3 0.3

O. 1 O. 1 0.7 0.1

- 800 - 800 800 - 1000

where P is the applied load. Specimen compliance was monitored by measuring the strain on the back surface of the specimen or by the Crack Opening Displacement (COD). These data can be correlated to crack length and recorded continuously by a Computer Aided Testing (CAT) program [8].

2.2. Environmental conditions Those tests performed in air correspond to a frequency of 10 Hz and mean stress ratio of R = ~rmi,/O'max = 0.1 where ~rma× and groin are, respectively, the maximum and minimum applied stress. Listed in Table 3 are the four different combinations of frequency, R-ratio and electrochemical potential. The air saturated artificial seawater was prepared in accordance with the A S T M Dl141-80 specification; its electrochemical potential was controlled by a potentiostat. A continuous flow was maintained at a temperature of 20 _+ 2 ° C such that it engulfs the specimen.

3. Test results As mentioned earlier, test data are obtained for the base material and weldment in air and seawater environment (Fig. 2).

1E-02,

1E-01

BASE

%-

MATERIAL

1E-03

1E-OZ Seawater

,**~ m + /'"

o o

~x x ~ / ' "

IE-04

air

× q?"

data IE-03

j

Z x3

000

IE-05

~ d~ 0

+ Air

%~ IE-06

a

1E-04

o b

×c 1E-05 100

10

A K ( MPa ~ m ) Fig. 2. C r a c k growth d a t a for base m a t e r i a l .

138

L. Bertini / Fatigue crack growth in C=Mn steel weld /oints

160

JB A S E

140

MATERIAL

IE-02

j/

IE-OI

t ,.A.z. i

Type "b" T e ~ "

120

~.

1E-02

XXXNxX ~x

1E--O3

¢9 O

lOO

o o 80 .~ m

J+ Seawater ~ l~a.o~

1E-03

60

z

X

Fit on ,ir data

40

~ +/

o,

~-----~

, ' ~

o

o~

'~ Ig-05

1E-04

20 0

I

0

I

2

I

Q

I

4 6 8 Load (KN) Fig. 3. Fatigue cycleswith crack closure effect.

i0

1E-06

d

/ ,

10 AK

, I 1E-05

100

( MPa 7-fia- )

Fig. 4. Data on crack growtli in HAZ,

3.1. Base material specimens Corrosion fatigue crack growth rate data are obtained in air and for the four diffcrent type of tests specified in Table 3. The results are displayed in a plot of d a / d N versus AK which is proportional to AP = Pma×- Pmin and can be computed from the standard relationship given in ASTM E647-81. Their trends are typical for the C - M n structural steels in seawater [9]. Crack growth rates tbr the Type a and b tests for AK in the intermediate and high ranges (greater than 15 MPa(n~) arc enhanced by the seawater in comparison with that in the air environment. This effect is increased slightly as the frequency is lowered from 1.0 to 0.3 Hz. Elevating the R ratio from 0.1 to 0.7 a n d / o r lowering the electrochemical potential to - 1 0 0 0 mV Ag/AgCI result in effects similar to those observed during Stress Corrosion Cracking (SCC). The relatively low d a / d N values in Type b tests for AK < 15 MPav/m could be attributed to "crack closure" as a result of oxide wedging [10]. This effect can be identified with changes in the slope of the Back Face Strain (BFS) versus the applied load curve in Fig. 3. The other test types did not reveal this phenomenon.

3.2. Weldjoint specimens Displayed in Figs. 4 and 5 are, respectively, the d a / d N versus AK data pertaining to crack growing in the heat affected and fusion zone. The general trends appear to be similar to those for the base material. A direct comparison of the d a / d N data for base material and H A Z can be found in Fig. 6. Noticeable difference is observed for low values of AK; the curves converge into one another as AK is increased. This is in agreement with the earlier findings [11,12] for fatigue crack growth in welds made of C and C-Mn structural steels tested in air. Crack closure effect were also observed for growth in the H A Z as shown in Fig. 3. This is attributed to the residual stresses produced in the H A Z as a result of welding. A knowledge of the reduction in the effective stress intensity factor range is thus required to quantify the crack growth data.

L. Bertini / Fatigue crack growth in C-Mn steel weld joints IE-02

1E-02

IE-OI

lAIR TESTSi

00 IE-03

139

Seawater

1E-02

.-.IE-03

°

R0

,4

IE-04

o

°

0

e~+~

Fit on

IZ-Oa

o

air data ~,/,,, F

~Z

:

~ 1E-04 Z

"~ I N - 0 5

0

+ Base Mat. o H.A.Z.

1E-06 10

,

o

.

.

.

.

.

ZXK( MPa ¢ ~

.

1E-05 100

.

IE-06

i00

I0

A K ( MPa ~ -

)

Fig. 5. Data on crack growth in fusion zone.

)

Fig, 6. Comparison of data tbr crack growth in base material and HAZ in air environment.

4. Residual stress analysis An empirical procedure is applied to find the residual stresses. This is based on strain measurements and residual stress relaxation with influence coefficients relations obtained from finite elements [13]. 4.1. Strain measurements

Illustrated in Fig. l(b) are the locations of strain gages attached to the specimen along a line normal to the plane of crack growth. Assuming that stress relaxation is complete, the linear theory of elasticity yields an expression for the x-component strain (E~) i recorded by the ith strain gauge:

1 (E~M), = ~ - [ ( o ~ ) i -

u(~y),]

(2)

where E and u are, respectively, the Young's modulus and Poisson's ratio. The original residual stress components are (~rx)i and (cry)i with x directed along the crack plane and y being normal to it.

Yi

II I IJ •

v

YJ

--

P'i

r j-I

Ilillllllllll Ill[lllllllll I[ll[llrlrr IIIllllllll I1[11111111[I [llillllllill

i,,,r,,rrrl ,r, tl[ a

P

Detail of crack tip mesh for K evaluation

r

Fig. 7. Assumed initial strain distribution in finite element model of compact tension specimen.

L. Bertini / Fatigue crack growth in C-Mn steel weld joints

140

4.2. Finite element analysis The compact tension specimen is discretized by using 251) plane stress four (4) nodes quadrilateral elements as shown in Fig. 7. This yields 500 degrees of freedom. Only one-half of the specimen needs to be analyzed because of symmetry. Linear elasticity is assumed for the material behavior. The residual stress field is assumed to be produced by an initial strain e~(~,) that could be approximated by a piecewise linear function with k intervals: for =

,

~)

Referring to Fig. 7, y~ and y] locates the upper and lower nodal positions for thc jth interval whose amplitude is chosen in accordance with the anticipated variation of the residual stresses, tnitial strains due to welding can thus be defined over k - 1 intervals such that one arbitrary interval can be chosen as the reference point. The stress and strain distribution is found by assuming a unit initial x-component strain in one of the intervals while the others arc zero. With reference to a set of N points located in the specimen, an influence coefficient re(!'),; corresponding to the x-component strain can be defined such that it represents the effect at the ith point due to the influence of a unit initial strain applied to the jth interval, i.e., 1

For a generic initial strain state as given by eq. (3), linear superposition can be applied to yield the strain at the ith point: k-

I

i

I

Let the N points coincide with the positions of the strain gauges; (e,)~ in eq. (5) may be equated with the measured strains (e~) i to yield a system of N linear equations: k-I

(e~M)i= 12 (E*)/i(E',~)i. i

i = 1.2 . . . . . N

i6)

1

solving for the unknowns (e,'Y)j. With k = N + 1, eqs. (6) can be solved for ( ~.w, )i and the resulting initial strain distribution is shown in Fig. 8. Using this distribution as the initial state of the finite element calculation, residual stress component cry along the path of prospective crack growth is obtained and shown in Fig. 9. According to this result, initial crack growth takes place in a compressive residual stress field.

4.3. Effectiue stress intensity factor The presence of residual stresses as those shown in Fig. 9 would affect the intensity of the crack tip stress field. Suppose that Knom denotes the nominal value of K obtained only by considering the applied load P. The influence of residual stress would change K ..... to Keff referred to as the effective K. The difference between K n o m and K~ff could be significant in fatigue because of crack closure [14]. At the maximum applied load, the corresponding K-factors may be related as Keftr~x =

.... K.om

+

K res m~,~

( - t

L. Bertini / Fatigue crack growth in C Mn steel weld joints

0.0015

141 150

15

--Kop (talc.) i t~--e--c c, c c-o---o lO I

o Kop (Exp.)

I

-~ Kres

I

5

•~ 0 . 0 0 1 0

100 50

/J

"-...

Z~

5q o

~

/ 0.0005

5

0

/'/

/'//

10

'\

J

50 .,.a U3 -100

/

0.0000

0

10 20 Distance from

30 weld

40

-15

-150 10

20

30 x,a

c e n t e r l i n e (ram)

40 ( mm )

50

60

Fig. 9. Normal stress and stress intensity factor variations along path of crack growth.

Fig. 8. Calculated initial strain distribution.

where K r e s corresponds to that introduced by residual stress effect. As the external load is decreased, the crack surfaces may come in contact [13]; let Pop denote the minimum load required to completely open the crack and Kop the value of K,o m at Pop. Crack closure will be observed within a fatigue cycle if rain Kop is greater than the minimum nominal value of K, K.o m. Assuming that K does not vary significantly as the external load is decreased from Pop to Pm~,,[13], an approximate evaluation of the minimum effective value of K can be written as Ken~ n =

M a x [ K o p , K n rain om] +Kre s

(8)

From eqs. (7) and (8), the following quantities may be obtained: AKeff

max

= Kef f

-

min K~fm~"= K.m~- Max[ Kop, K ...... ]

Neff = Serfmi. . . . . / g e f

The quantity

Kop may

_ (max[

(9)

S o p , Kn(%im n ] q'- K r e s ) / (

K ..... max _}_ S r e s ) .

be calculated from Pop, whose value may in turn be obtained from the condition:

min[Pop'Vext(X ) + Vres(X)] = 0 ,

(10)

O~x
50

0.5 --- ~ Keff ,~ K n o m o Reff

40

/

~-R ~ / / / / / / "

,/"

/

0.4

./'///" o

0.3 t~

p

////

v 20

/ 0.2

10

0.1

Air ' r e s t s 0

0

10

15 20 25 Crack length ( mm

30 )

35

Fig. 10. Variation of the effective and nominal values of stress intensity factor range with crack length.

L. Bertini / Fatigue crack growth in C-Mn steel weldjoints

142

w h e r e Uext(X) a n d V~e.~(X) are the crack surface d i s p l a c e m e n t d u e to a unit e x t e r n a l l o a d a n d to residual stresses respectively. T h e c a l c u l a t e d value of Kop a r e shown to a g r e e well with those o b t a i n e d e x p e r i m e n t a l l y as given in Fig. 9. T h e value o f K ~ was o b t a i n e d [15] by inserting in the m e s h e m p l o y e d for residual stress e v a l u a t i o n a z o n e with a r a d i a l a r r a n g e m e n t of e l e m e n t s s u r r o u n d i n g the crack tip (Fig. 7). T h e n , the initial strain d i s t r i b u t i o n was i n t r o d u c e d in the finite e l e m e n t m o d e l a n d the J - i n t e g r a l was c a l c u l a t e d on a s e m i - c i r c u l a r p a t h a r o u n d the c r a c k tip, which was s e l e c t e d a m o n g a few a l t e r n a t i v e p a t h s by a p r e l i m i n a r y c a l i b r a t i o n study. As a final step, K ~ was e v a l u a t e d from t h e ,/-integral value by the w e l l - k n o w n r e l a t i o n s h i p given by the L i n e a r Elastic F r a c t u r e M e c h a n i c s theory. F i g u r e 10 shows the c a l c u l a t e d value o f AK~ff, A K ..... R~n a n d R,,om as a function o t crack length for a typical F C G R test. T h e i n t e r a c t i o n o f r e s i d u a l stresses with R - r a t i o is not a p p r e c i a b l e , while the d i f f e r e n c e b e t w e e n n o m i n a l a n d effective value o f A K is l a r g e r in the initial p a r t of the test.

5. Fatigue crack growth rates

Q u a n t i t a t i v e a s s e s s m e n t s of t h e fatigue crack growth rates in the h e a t a f f e c t e d and fusion zone are m a d e by a p p l i c a t i o n o f the r e l a t i o n da -

dN

Ea, c,(aK)"'

it ~)

in which

6i=

1, O,

for k K i <~k K < k K i +! for A K , + , < A K < A K ,

In eq. (11), d a / d N is m e a s u r e d in m m / c y c l e a n d v a l u e s of the p a r a m e t e r s C i a n d n, t o g e t h e r with A K , can b e f o u n d in T a b l e 4. M e a n stress effect is not a c c o u n t e d for in eq. (11) b u t its influence is small for the A K r a n g e c o n s i d e r e d as e v i d e n c e d by the results in Fig. 10. C r a c k growth rates for T y p e b tests w h e r e s e a w a t e r plays a role can also be e s t i m a t e d from eq. (t 1) by using AK~f: to c o r r e c t for the oxide w e d g i n g effect.

Table 4 Fatigue parameters and stress intensity t'actor ranges in base material Type

(7i

air

1.288×10 I() 9.333×10 s

4.32 2.20

15.0 25.;

a

4.169× 10 s 3.090× 10 s

3.0{~ 2.60

[5.7 2"3

b

2.692X 10 u() 3.388 x 10 > 1.047×10 "

4.3'0 0.4'47 2.37

i l.0 203 217

c

3.981×10 la 2.042× 10 ~

8.1 C) 0.653

il.I~ 144

d

1.995X10 2.

17.6 1.06 1.68

15,7 18.2 23.,1

1.380x 10 5 1.950× 10 ~

n,

J K, (MPav"m }

143

L. Bertini / Fatigue crack growth in C Mn steel weld joints IE-02

] /" I ~ ,,"" ~ Typical Base Mat. Scatter Band (2 S D ~ , , - ' "

1E-02

"3

..

"~

[ I ,," / Typical Base Mat. ~ , , , / / , Scatter Band (2 SD) /7/~,,,,,"

_zl/~

o

IE-03

IE-03 ~

"""

O0

r~

//'"

[.Az

///'/

- Knom i

1E-05

IE-05

IE-04

, ~

1E-05

IE-02

IE-03

~ Keff

IE-05

Observed FCGR (mm/cycle)

IE-O2

IE-03

1E-04

Observed FCGR (mm/cyele)

Fig. iI. Fatigue crack growth rates in HAZ correlated by nominal stress intensity factor range.

Fig. 12. Fatigue crack growth rates in }|AZ correlated by effective stress intensity range.

Predictions m a d e by application of eq. (11) for the weld s p e c i m e n s are shown in Figs. 11 to 14 inclusive w h e r e both AK¢f~ and A K . . . . were used and they are c o m p a r e d with the experimental data. Figure 11 shows that the predictions based on A K , o m for the H A Z are o v e r e s t i m a t e d by as much as o n e order of magnitude, particularly for low d a / d N values. Much better a g r e e m e n t is obtained for AK~ff given in Fig. 12. This validates the correction introduced for the residual stress field at least for the problem considered. The s a m e conclusion can be drawn from Figs. 13 and 14 for the fusion z o n e where better correlation with experimental data is obtained w h e n using AK~ff in Fig. 14 in contrast to A K ..... in Fig. 13.

1E-02

I Typical Base Mat. S c a t t e r Band (2 SD)

]

____~/"

,,//'/

//

IE-02

Typical Base Mat. "/ S c a t t e r Band (2 SD) ~----->"" /

./ O O

o

IE-03

,,"~

~ iE-oa

./0

r~ rJ

[.9

o,~" //'/'/

, /'/"

,,.,/"

IE-04-

Z

.2 ~ F u s i o n 1E-05 1E-05

1E-04

Zone -

Knorn

IE-03

a~ 1E-02

O b s e r v e d FCGR ( r a m / c y c l e ) Fig. 13. Fatigue crack growth rates in fusion zone correlated by nominal stress intensity factor range.

Zone 1E-05 "/ 1E-05

Keff

I 1E-04

IE-03

1E-02

O b s e r v e d FCGR ( m r n / c y c l e ) Fig. 14. Fatigue crack growth rates in fusion zone correlated by effective stress intensity factor range.

144

L. Bertini / Fatigue crack.growth in C--Mn steel weld joints

6. Concluding remarks W h a t has b e e n shown is that crack growth b e h a v i o r in w e l d m e n t s is not only a f f e c t e d by e n v i r o n m e n t such as s e a w a t e r but also by residual stresses as a result o f the w e l d i n g process. R e c o g n i z i n g the fact that the l a b o r a t o r y test s p e c i m e n data could differ significantly f r o m t h o se in actual structural m e m b e r s , the p r e s e n t investigation has led to the following conclusions: • A s s e s s m e n t o f t h e c o r r o s i o n f a ti g u e crack growth data for the C - M n structural steel a g r e e d both qualitatively an d q u a n t i t a t i v e l y with e x p e r i m e n t s ; t h ei r t r e n d s are similar to o t h e r types o f steel in seawater environment. • Cr ack g r o w t h in weld j o i n t s p e c i m e n s w e r e f o u n d to be lower than that of t h e base m a t e r i a l for low A K range. This a p p l i e d to both the air and s e a w a t e r e n v i r o n m e n t . • A n e m p i r i c a l p r o c e d u r e involving e x p e r i m e n t a l and n u m e r i c a l w o r k was used successfully for d e t e r m i n i n g t h e residual stresses in weld specimens. • C o r r e c t i o n d u e to residual stress effect w e r e significant an d the use of AK~,~ in relation with crack c l o s u re led to b e t t e r c o r r e l a t i o n b e t w e e n analytical and test results.

Acknowledgement T h e financial s u p p o r t s p r o v i d e d by S n a m p r o g e t t i in Milan and T e c n o m a r e in V e n i c e are gratefully acknowledged.

References [1] H. Baumgardt, H. DeBoer and B. Miisgen, Special features in the development of high strength steels for offshore technology~ 2nd Int. Conf. on Offshore Welded Structures, London, Paper 46, 1982. [2] I.S. Cole, R. Brook and I.C. Howard, The environmental performance of higher strength steels for offshore applications, Enrironment Assisted Fatigue, EGFT, P. Scott, ed.. Mech. Eng. Publ.. London, (1990) pp. 353- 366. [3] B. Tomkins, Coping with corrosion fatigue in design with particular reference to pressure vessels and offshore structures. Proc Corrosion-Fatigue USSR-UK Seminar, Lvov (May 1980) pp. 135-147. [4] S. Dharmavasan and W.D. Dover, Non destructive evaluation of offshore structures using fractme mechanics, DOC NDE/87/A/20, University College, London, 1987. [5] L. Bertini, F. Citernesi and F. Trave, Corrosion fatigue crack growth of welded joints in marine environment. Proc. X ~ National Conf ofA.LA.S., ETS ed., Pisa, Italy (1087) pp. 717-731 (in Italian). [6] L. Bertini, Corrosion fatigue crack growth rate in a C-Mn steel and in its welded joints in seawater. Enrironment Assisted Fatigue, EGF7, P. Scott, ed., Mech. Eng. Publ., London, (1990) pp. 45-60. [7] Det Norske Veritas, Rules for the design, construction and inspection of offshore structures, 1977. [8l L. Bertini and E. Vitale, A multistation computer aided material testing system for fracture mechanics and fatigue, Proc. ASME Int. Comp. in Eng. Conf. and Exhib., New York (1987) pp. 23-29. [9] P.M. Scott, The effects of seawater on corrosion-fatigue of structural steels, Proc. Corrosion-Fatigue USSR-UK Seminar, Lvov (May 1980) pp. 135-147. [10] R. Murakami and W.G. Ferguson, The effects of cathodic potential and calcareous deposits on corrosion fatigue crack growth rate in seawater for two offshore structural steels, Fatigue Fract. Eng. Mater. Struc. 9 (6) (1987) 477-488 [11] D. Benoit, H.P. Lieurade and M. Truchon, A study on the propagation of fatigue cracks in the HAZ of welded joints in E36 steel, European Offshore Steel Research Seminar, Cambridge, UK, 1978. [12] D.A. Davis and E.J. Czyryca, Corrosion fatigue crack growth characteristics of several HY-100 steel weldments with cathodic protection, Corrosion Fatigue: Mech. Met. Electrochem. and Eng. (ASTM STP 80i: Philadelphia, 1983) pp. 175-196. [13] L. Bertini, A quantitative assessment of the residual stress effects on fatigue crack propagation in structural steel weldments in air and seawater, Proc. 2nd Int. Conf. on Fat. and Stress (I.I.T.T. Int.: Paris, 1989)pp. 202-211. [14] M. Beghini and L. Bertini, Fatigue crack propagation through residual stress fields with closure phenomena, [£ng. Fract. Mech. 3 (3) (1990) 379-387. [15] M. Beghini and L Bertini, Analytical and numerical evaluation of the residual stress effects on fatigue crack propagation, Proc. 4th Int. Conf. on Comp. Meth. and Exp. Meas. (Springer: Berlin, 1989) pp. 187-198.