An Analytical Model for Fatigue Life Prediction Based on Fracture Mechanics and Crack Closure (1996)

ELSEVIER

J. Construct. Steel Res. Vol. 37, No. 3, pp. 229-261, 1996 Copyright © 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved S0143-974X(96)00006-$ 0143-974X/96 $15.00 + 0.00

An Analytical Model for Fatigue Life Prediction Based on Fracture Mechanics and Crack Closure

J. B. Ibs¢ & H. Agerskov

Department of Structural Engineering, Technical University of Denmark, Building 118, DK- 2800 Lyngby, Denmark

(Received 15 June 1995; revised version received 29 October 1995; accepted 18 January 1996)

ABSTRACT

Fatigue in steel structures subjected to stochastic loading is studied. Of spe- cial interest is the problem of fatigue damage accumulation and in this con- nection, a comparison between experimental results and results obtained using fracture mechanics. Fatigue test results obtained for welded plate test specimens are compared with fatigue life predictions using a fracture mechanics approach. In the calculation of the fatigue life, the influence of the welding residual stresses and crack closure on the fatigue crack growth is considered. A description of the crack closure model for analytical determination of the fatigue life is included. Furthermore, the results obtained in studies of the various parameters that have an influence on the fatigue life, are given. A very good agreement between experimental and analytical results is obtained, when the crack closure model is used in determination of the analytical fatigue lives. Both the analytical and experimental results obtained show that the Miner rule may give quite unconservative predictions of the fatigue life for the types of stochastic loading studied. Copyright © 1996 Elsevier Science Ltd.

C

C Ci

C

d dc

dn

NOTATION

Half length of physical crack; crack depth Crack growth coefficient Initial crack size

Crack shape parameter

Half length of effective crack

Crack growth rate

229

230 J. B. Ibsa, H. Agerskov

f

f . Fs, F ,FT, Fo g Zi

m

R

S So t V Wi Ac* AK AKeff AKeff, th AS A&ff or

ar tro

P OL CA VA max rain

Stress intensity factor correction; influence function for displacements Yield stress Ultimate tensile strength Stress intensity factor corrections Influence function for displacements Length of bar element i (amount of permanently deformed material) Crack growth coefficient

Snain Stress ratio, R -

Smax Remote applied stress Crack opening stress Thickness of specimen Crack surface displacement Half width of bar element i Crack extension increment Stress intensity factor range Effective stress intensity factor range Effective threshold stress intensity factor range Remote applied stress range Effective stress range Normal stress Residual stress Flow stress, O'o = ½(fy +fu) Plastic zone size Constraint factor on tensile yielding Constant amplitude Variable amplitude Maximum Minimum

1 INTRODUCTION

Fatigue crack growth under variable amplitude loading was first studied for aircraft structures. Later on, variable amplitude fatigue was also found to be relevant to bridges and various structures in industrial fields, such as ships, cars and offshore structures.

A considerable number of experimental investigations on variable amplitude fatigue has been carded out over the years, and on the basis of the results obtained in these investigations, various empirical formulae for fatigue life

Fatigue life prediction 231

predictions have been proposed. However, in order to attain a better physical understanding of fatigue crack growth under various load histories and different stress states, theoretical analyses are needed. Theoretical models which consider the material behaviour and the mechanistic aspects of fatigue crack growth, such as crack closure and load interaction effects may provide the basis for a better understanding of different fatigue crack growth behaviours. Fatigue test results from laboratory experiments can provide important and useful information, however the results may only be valid for the specific test conditions applied, and often they may not be readily generalized. On the other hand, a theoretical model, which is based on a solid background and checked against fatigue test data for various test conditions may provide a reliable basis for the analysis of fatigue crack growth cases, for which no pre- knowledge exists.

During the last 5 years, an investigation on fatigue life of offshore steel structures has been carried through at the Department of Structural Engineer- ing of the Technical University of Denmark. This investigation comprises mainly experimental work. The present study is a continuation of this investigation, however, it concentrates on the theoretical determination of the fatigue life. Comparisons between theoretical and experimental results are included.

2 CRACK CLOSURE

In the following, the significance of crack closure for fatigue crack growth is investigated. The crack closure model accounts for load interaction effects, such as retardation and acceleration, under variable amplitude loading.

In a body with crack growth, the crack surfaces remain closed during a part of the load cycle even under tension-tension loading. Elber 1 was the first to notice this phenomenon and related an effective stress intensity factor range for a fully open crack to the fatigue crack growth rate.

The mechanisms of crack closure have been attributed to plasticity-induced closure, roughness-induced closure and environment-induced closure. In the present investigation, only plasticity-induced crack closure is considered.

Plasticity-induced crack closure is caused by residual plastic deformations in the wake of an advancing crack. During the unloading part of a load cycle, the crack will close (at least partly) before the minimum load level is reached due to the residual plastic elongations, which are left along the crack surfaces as the crack grows. After full unloading, compressive residual stresses will be present in the wake of and ahead of the crack tip.

As a result of the closure of the crack during a part of the applied loading range, the driving force for fatigue crack growth is the effective value AKeff, which is the part of the total stress intensity factor range, AK where the crack is fully open:


a K o . = Kmax - (1)

where Kop is the stress intensity factor representing the value at which the crack is fully open, i.e. when there is no longer crack surface contact behind the physical crack tip during the uploading part of a load cycle.

If effective stress intensity factor ranges are considered, the fatigue crack growth rate in structural steels and steel weld metals may be presumed to follow a power law of the following form:

dc = C ( a K ~ f - AKe~n}f, th )

for AKeff ->AK~ff,~ (2)

dc - - = 0 dn

for AKeff <AK~ff, th

where AKeff, th is the effective threshold stress intensity factor range, below which no crack growth takes place, considering the effect of crack closure. The crack growth rate eqn (2) is based on effective stress intensity factor ranges for a fully open crack. Therefore, when used in this equation, the crack growth coefficients C and m can be considered as constants for a given material. The Paris-Erdogan crack growth equation 2,3 is often used for crack propagation calculations, using full stress intensity factor ranges, AK. How- ever, the Paris-Erdogan equation has been found not to correlate with fatigue crack growth rates in the near threshold region, neither does it account for mean stress effects. Furthermore, the Paris-Erdogan equation has been shown not to correlate with fatigue crack growth rates of small cracks. Equation (2) takes account of the near threshold fatigue crack growth through the term, AKefe, th. Furthermore, the effect of the mean stress on the fatigue crack growth is taken into account in eqn (2), since it is based on effective stress intensity factor ranges for a fully open crack. In general, the effective stress intensity factor range of a stress cycle depends on the stress ratio and the maximum stress level in the stress cycle. If crack closure can be determined for small cracks, there is some evidence 4'5 that modifying the driving force into AKoff, this will result in good predictions of fatigue crack growth of small cracks.

2.1 Crack closure models

In the present study, a strip fatigue crack growth analysis model based on the Dugdale6-Barenblatt 7 strip yielding assumption is used.


Compared to the Dugdale-Barenblatt procedure, the present model is modified to leave plastically deformed material along the crack surfaces as the crack advances. Plane stress and plane strain conditions, as well as conditions between these two, are simulated in the model by using a constraint factor, a on tensile yielding at the crack front to approximately account for three- dimensional stress states. 8

The crack closure model presented herein will be used for calculation of the fatigue crack growth life of welded plate elements under constant amplitude loading and under stochastic loading. Methods to include the influence of stress concentrations and welding residual stresses upon crack closure and fatigue crack growth in the analysis are presented.

In the following, a crack closure mechanism-related and generally appli- cable model is described. The basic layout of the strip yielding model was established in the early works of Dill and Saff 9 and Fuehring and Seeger. 1° Newman 8 improved the strip model by introducing a relatively sophisticated numerical procedure which led to better results for the crack tip plasticity computations for variable amplitude loading. Wang and Blom 5 developed a new procedure based on the weight function theory in order to facilitate the analysis of any two-dimensional geometry.

2.2 Analytical crack closure model

In the Dugdale model, 6 yielding is assumed to occur along a strip ahead of the physical crack tip. Crack tip plasticity is corrected in the Dugdale model by adding the plastic zone size, p to the physical crack length. This defines the fictitious or imaginary crack. The Dugdale model states that there will be no stress singularity physically existing at the fictitious crack tip, i.e. the stress at the fictitious crack tip should not be infinite. Thus, the stress intensity factor, K% due to the uniform tension in the plastic zone will numerically be equal to the stress intensity factor, Ks due to the remote load, i.e.

Ks + K ro = o. (3)

The Dugdale plastic zone size, PDug can be determined by solving eqn (3). In the present crack closure model, rigid-perfectly plastic material behav-

iour is assumed. For the majority of metallic materials, there is more or less strain hardening in the plastic deformations. As a first approximation for strain hardening, a flow stress tro, which is the average value of the yield stress, fy and the ultimate tensile strength, fu is proposed in the model. The crack surface displacements, which are used to calculate contact stresses during unloading, are influenced by plastic yielding (and reversed plastic yielding) at the crack tip and residual deformations left in the wake of the advancing crack (see


S

_JJ I._

I Y

2d

%" 2C "~p . ~ X

2Vt

S

Fig. 1. Centre crack tension specimen with Dugdale plastic zones and residual plastic deformations.

Fig. 1). Upon reloading, the remote applied stress level at which the crack surfaces become fully open (no crack surface contact) is directly related to the contact stresses. This stress is called the crack opening stress and it is denoted So.

The crack surface displacements are obtained by superposition of the sol- utions for two elastic problems, a crack subjected to remote applied load and a crack subjected to a uniform stress applied over a segment of the crack surface (see Fig. 2). The uniform stress applied over a segment of the crack surface represents the contact stress for an element along the crack surface or the residual stress for an element in the plastic zone ahead of the physical crack tip. The crack displacements of the effective crack surface, that is, the

Y S

_[_LJ.J_ IJ L

17 rill

I. d I

Fig. 2. Crack surface displacements of effective crack.


crack surface displacements of a crack consisting of the physical crack and the plastic zone ahead of the physical crack tip, V(x) (see Fig. 2) at any point x will be determined by the relation

V ( x ) = S f ( x ) - o 'g (x ,Xo) . (4)

where Sf(x) is the crack surface displacement at x due to the remote applied stress S, and og(X,Xo) is the crack surface displacement at x due to a uniform stress o- acting on a segment of the crack surface with the centre at Xo (see Fig. 2).

If the effective crack is divided into a number of constant stress bar elements, and all the elements along the effective crack are considered, the crack surface displacement at the centre of element i can be calculated from

n

g i = S f (x i ) - Z o'jg(xi'xj) j = l

(5)

where n is the number of elements, o-j is the stress on element j, S is the remote applied stress, f (x3 is the crack surface displacement at element i due to unit remote applied stress, and g(xi,Xj) is the displacement at element i due to unit stress acting on e lement j . Equations for the calculation of the influence functions f(xi) and g(xi,xj) for a crack in an infinite plate are given in Ref. 11. These equations have been modified in Ref. 12 for the calculation of the influence functions for a surface semi-elliptical crack in a structural detail.

2.3 Numerical aspects

Equation (5) can be rewritten in a short form in the following way:

n

ojg o = Sfi - Li (6) j = l

where Li is the length of element i. For elements in the plastic zone and elements along the physical crack surface, which are in contact, L / i s equal to the displacement at element i of the effective crack surface (Vi = L3. The length of a bar element, Li is the amount of permanently deformed material. For elements along the physical crack surface, which are not in contact, the crack surface displacement Vi is greater than Li and oi = 0 for these elements. The plastic deformation, Li only changes, when an element yields in tension (oi >--oto',,) or compression (O" i ~ - - O ' o ) , assuming rigid-perfectly plastic

236 J. B. Ibs~, H. Agerskov

material behaviour. In eqn (6), functions f and g~j are based on the effective crack length, d.

To keep the number of elements to a reasonable size (30-60), a lumping procedure is used. This procedure combines adjacent elements ( i -1 and i) to form a single element. In the present modelling, the following criterion of lumping is used:

2(w~ + W i _ l ) ~ c - - Xi-- 1. (7)

In eqn (7), w i is half the width of element i, x i _ 1 is the coordinate location for element i - 1 , and c is half the physical crack length (see Fig. 3). If the criterion in eqn (7) is fulfilled, the elements i - 1 and i are lumped together to form a single element. Lumping is carried out starting with the elements farthest away from the physical crack tip. The consequence of this sequence of combining adjacent elements and the criterion in eqn (7) is that the elements far away from the crack tip are more likely to be lumped together as compared to those elements nearer to the crack tip. In the present modelling, lumping is carded out in a way so that the number of elements is always greater than 30 (except in the very initial stage of the crack growth simulation). In the lumping procedure, the width of the lumped element is the sum of the widths for the two adjacent elements, and the length is the weighted average of the two:

~mta

1..t 1.1 k l 1 1

I Element n Crack tip / Element i el? mcnt (i=11)

Ln~ .~ent 1

I 'q-~

w ~,

X

Fig. 3. Schematic of analytical crack closure model at minimum applied stress.


L i _ l W i _ 1 .t- L i w i ( L i _ 1)ne w = (8 )

Wi-- 1 q" Wi

Crack extension is simulated at the moment of maximum applied stress. The amount of crack extension, Ac* is chosen as

Ac* = 0.01 Pmax (9)

where Pm,x is the size of the plastic zone caused by the maximum applied stress during the Ac* growth increment. The crack growth increment given in eqn (9) will result in reasonable computation times and good prediction accuracy on the calculated results. 5,13 In the simulation, the crack opening stress level is held constant within each crack extension increment, Ac*.

2.4 Effective stress intensity factor ranges

The effective stress intensity factor range, AK~ff may be expressed as

AKeff = z l S e f f f ~ - ~ ( 1 0 )

where f is a dimensionless stress intensity factor, AS~ff is the effective stress range considering crack closure effects, and c is half the physical crack length.

For constant amplitude loading, AS~ff is determined as

ASef f ~-~ S m a x - Smi n if Stain ----- So (11)

AS~ff = Sm~ - So if Smi n < S o. (12)

For variable amplitude loading, AS~ff for a stress cycle with a minimum stress, S m i n and a maximum stress, Smax just following it, is determined as follows 8

I f : Smi n ~--- S o

then: AS~ff = Sm.x - So

If : Stain > So

then : AS,el = Sm,x - Stain

o r " ASeff = [(Sma x - So) m - (Smi n - - So) ,]u m.

(13)

(14)

(15)

(16)

(17)

The use of eqn (17) is necessary because no crack growth law, when expressed in terms of a power function with an exponent m =4= 1 would sum to the correct growth increment under variable amplitude loading. Equation (17) is only


applied, when Smi n is greater than So, and only when the current maximum stress is higher than the highest maximum stress occurring since a stress excur- sion crossed So. Otherwise, when Smin is greater than So, eqn (16) is applied. For instance, if Smin,k < So < Smax,k for stress cycle No. k, and the load excur- sion S~nax,k t o Smin,k+ 1 is extremely small and Smin,k+~ is assumed to be greater than So, then the growth increment (Ack + Ack+~) should be equal to the growth increment of one stress cycle of range (S~nax,k+~- So), assuming S . . . . k+l ;> Smax,k+l" On the other hand, if Smin,k+ 1 is less than So, then the growth increment Acj,+l alone should be equal to the growth increment of one stress cycle of range (Smax,k+~ -- So). Equations (13)-(17) take account of these limit- ing behaviours.

2.5 Crack opening stresses

Different methods for calculating the crack opening stress level, So may be used. 4'8 In the present modelling, the crack opening stresses are calculated by a displacement method. The crack opening stress is determined by setting the displacement at each physical crack surface element due to an applied load increment (So - Smi,) equal to the displacement at the corresponding element due to the contact stresses at Smin. The maximum value of (So - & i n ) gives the load level at which the last element in contact separates.

The displacement at element i along the physical crack surface due to a remote applied stress increment (So - Smin) is

VSi _~. ( S o - S m i n ) f ( x i ) . (18)

The displacement at element i due to the contact stresses o) is (see Fig. 3):

n

V~ = - Z °rjg(xi,xj)" j = l l

(19)

Equating eqns (18) and (19) and solving for (So)i gives

_ ~ %.g(xi,xj) (So), = Sn n j="7, f (xi) " (20)

The maximum value of (So),. gives the crack opening stress, So. The functions f(xi) and g(xi,x) in eqns (18-20) are based on the physical crack length, c.


2.6 Proof testing of crack growth program

Based on the crack closure model described above, a computer program FAWS (Fatigue Analysis of Welded Structures) has been developed, lz In Ref. 12, analytical results obtained by use of the program for various fatigue crack growth cases are tested against both experimental results and crack closure analyses from the literature. In general, very good agreement was obtained. In the following, some examples are given showing comparisons between analytical results obtained from the program and test results.

The effect of a single tensile overload on the subsequent fatigue crack growth under constant amplitude loading is experimentally well documented. On the other hand, even in the simple case of a single tensile overload different well known models, such as the Wheeler model 14 and the Willenborg model 15 fail to model the delayed retardation following the single overload, see e.g. Ref. 16.

The crack closure model and the crack growth computer program for the plane strain condition are tested against the experimental results of ChananP 7 for the overload effect in an A1 7075-T73 aluminium alloy material. For the test specimen of thickness t = 12.7 mm in Ref. 17, the plane strain condition according 1-o the ASTM Method E399-74 was fulfilled for both the baseline CA loading and the overload. Single-edge notched (SEN) specimens of width 63.5 mm were used. The yield stress of the A1 7075-T73 material was reported as fy = 419 MPa and the ultimate tensile strength as fu = 491 MPa. The test results from Ref. 17 for a single overload are shown in Fig. 4 as triangular and square points. The triangular points were determined from surface crack length measurements using a travelling microscope, and the square points were determined from the striation spacing on the fracture surfaces measured by use of a scanning electron microscope (SEM). The prediction by the strip model is made with the introduction of Irwin's TM plane strain constraint factor a = 1.73, and the prediction is represented by the solid curve in Fig. 4. In this and in the following predictions, the fatigue crack growth rates are determined from the relation in eqn (2). As may be seen from the prediction given in Fig. 4, there is an immediate crack growth acceleration just after the appli- cation of the overload. Initial acceleration of the crack growth was also reported in Ref. 17. The initial acceleration is followed by retardation until a minimum value is reached. The crack growth then gradually recovers to the normal level as the crack grows out of the overload plastic zone. As may be seen in the comparison in Fig. 4, a good agreement exists between the test results and the analytical prediction under the plane strain condition.

Predictions in the plane stress condition are performed using the crack growth computer program on Jacoby et a l . ' s 19 experiments on the aluminium alloy A1 2024-T3. Thin centre crack tension (CCT) test specimens were used


P(kN)

~/NOL = l 35.6

dc/dn (mm/cy'cle) 17.8 ~ A A A / / A A A A ....

4 h kl 7075-T73 1.78~ I v , ! ! ! ' ' ' - Tixne

3

O- I [ a = 1.73 7. 8- 5- ~ Ac" = O.Ol p

4-

3-

z~ z5.9 se . l 26'.s aa'.5 z6'.? ~6'.0 ~'.1

Crack lenEt, h, c (mm)

Fig. 4. Variation in fatigue crack growth rate in AI 7075-T73 material following a single tensile overload.

(with thickness equal to 1 mm), and the amplitude of the applied load was relatively high. Under these conditions, the crack tip plane stress state can be assumed (c~ = 1) for all tests in Ref. 19. The width of the test specimens was 80 mm. The material properties of the A1 2024-T3 material was not given in Ref. 19, however, the yield stress is assumed to befy = 340 MPa and the ultimate tensile strength is assumed to be fu = 430 MPa based on the properties tabled in Ref. 16 for the same material.

The comparison of test results and analytical prediction for 40 overload cycles is shown in Fig. 5. The test results are represented by the solid symbols, and the solid curve represents the prediction by the crack closure model using FAWS. As shown in Fig. 5(a), the crack growth prediction by FAWS models the crack growth behaviour for 40 overload cycles quite well. For the case of multiple overloads, prediction can only be logically made with a crack growth model which involves the plastic deformation in the wake of the crack. In Fig. 5(b), the variation of the crack opening stress is shown as a function of the crack length. The overloads cause larger-than-stable plastic stretches left along the crack surface and in the plastic zone. When the stretches move to the crack surface, larger-than-stable contact stresses will appear on the crack surface, So will increase [see AB in Fig. 5(b)], and consequently the crack


dc/dn t |:

I , | ,

4 ,

8 .

| .

I - I -

4-

i .

%

80-

70-

60-

50-

40-

30-

20

( ~ / c y e l e ) s(m~a)

150

\ ~ 3 100

50

o

~ "Ih'~lieUon

~/NoL- 4O

SO

(MF'a)

1'1 l~ t'a {4 Crack lenlrtb , c (ram)

(a) B

A 150

/ ~ 100

~ ~ 1.0

A &c" = O.Ot p

s(m~a) jNOL= 40

' n l ~ I

1'0 1'1 l'z 1'a 1'4 Crack length, e (ram)

(b)

Fig. 5. Comparison of experimental and calculated crack growth rates following 40 tensile overloads. (a) Variation of crack growth rate. (b) Variation of crack opening stress, So.

growth is retarded. The crack opening stress level will recover to the stable level, when the overload plastic stretches move away from the crack tip [see BC in Fig. 5(b)].

Figure 6 shows a comparison between analytical prediction and test results for the crack growth behaviour for the multiple overload case with a single underload following immediately after the overloads. The test results are represented by the solid symbols and the analytical prediction is given by the solid curw~. As can be seen in Fig. 6, there is a very good agreement between


d c / d n (~/cycle) AI 2024-T3

0.1

1- ~ P r e d i c t i o n

/"- l A^^AA

Ac = O.Ol p -50

0.01 l t i 9 10 Ill 12 13

Crack length, c (ram)

/ NUI,~ 1

.- Time

Fig. 6. Comparison of experimental and calculated crack growth rates after multiple tensile overloads followed by a single compressive underload.

the prediction and the test results. In general, an underload following one overload or multiple overloads will have a significant influence on the fatigue crack growth retardation due to the overload(s). When large compression is applied, the plastic stretches both on the crack surface and ahead of the crack tip are 'squeezed' flat. The crack surface near the crack tip will then contact at a lower-than-normal level, and consquently the crack growth is accelerated.

3 FATIGUE LIFE PREDICTIONS FOR WELDED PLATE ELEMENTS

Fatigue tests on plate elements with groove welded transverse attachments have been carried out at the Department of Structural Engineering of the Tech- nical University of Denmark. 2°'21 In Refs 20 and 21, two types of test specimens were investigated (see Fig. 7), with a plate thickness of the main plate of 16 and 8 mm, respectively. The material used in these test series was St. 52-3 (Fe 510C) according to DIN 17100. The yield stress, fy, and the ultimate tensile strength, fu, of the material used were fy = 400 MPa and fu = 530 MPa for the main plate of thickness 8 mm, andfy = 405 MPa and f , = 575 MPa for the main plate of thickness 16 mm, respectively. These values Offy and f~


300

30C

- I --

90

I *_p

-.~.__ • PL 10x90

1.5°Ii

" " i -

187.5

167 -~

_11 12

" 9o t

187.5

I _*p

_.~.__ ,•L 5xB

-7 oII' 0 - '~-~ PL 8xB

B=/.O Meosures in mm

Fig. 7. Fatigue test specimens.

were determined from tension coupons taken from the plates that were used for the fabrication of the test specimens. 21 The tests of the present investigation were carried out under either constant amplitude loading or stochastic loading.

The fatigue crack propagation analysis is carded out by a fracture mechanics approach employing the crack closure model presented in Section 2. In the following, only the fatigue life corresponding to the crack propagation is calculated. The initiation or nucleation period of cracks growing from defects in welded connections is most often considered to be insignificant in relation to the crack propagation period. 22

3.1 Fracture mechanics analysis of welded plate elements

In the analysis, it is assumed that the fatigue crack propagates in a semi- elliptical shape, as shown in Fig. 8. This assumption is in good agreement with experimental observations of the crack shape in many of the fatigue tests carried out. Furthermore, in the analysis it is assumed that the fatigue life of

244 I. B. Ibso, H. Agerskov

t '

Fatigue f racture s u r f a c e

c : c r a c k d e p t h 2b: surface width of c r a c k

Fig. 8. Schematic of semi-elliptical surface crack growing from weld toe.

the welded plate test specimens can be estimated by considering only one crack growing from the weld toe as shown in Fig. 8. In some cases, this corresponds to the actual crack growth behaviour observed in the fatigue tests carried out. However, in other cases it was observed that two or more small cracks grew together, forming a single fiat crack at an early stage of the fatigue life. In the crack propagation analysis, crack coalescence at an early stage may be modelled in an approximate way by considering only one crack with a low value of the semi-axis ratio, c/b (see Fig. 8).

3.1.1 Stress intensity factor due to remote load A widely used approximative calculation of the stress intensity factor for a surface semi-elliptical crack in a structural detail is obtained by using the method given in Ref. 23. Using this method, K is expressed as follows

K = Fs.FE.FT.FG.S. (21)

where Fs is the (front) free surface correction factor, FE is the elliptical crack shape correction factor, Fv is the finite plate thickness (or width) correction factor, Fc is the stress gradient or geometry correction factor, S is the remote applied stress and c is the crack depth. The stress intensity factor corrections in eqn (21) may be found from Refs 11, 22 and 23:

C Fs = 1.12 - 0 . 1 2 . ~ (22)

C "] -- 1/2

FE = 1 + 4.5945(~)1"65J (23)

and


FT = x/sec('n'cl2t) (24)

in which t is the thickness of the specimen. Fo may be determined from the results of a finite element analysis using

the weight function method. In the present study, F~ is computed according to the method presented in Ref. 23, by integrating away the normal stresses in the uncracked body at the line, where the crack is to be inserted. F~ is high at small crack depths, c, which reflects the high stress concentration at the weld toe, but then at larger crack depths, F~ decays to a value of 1, or no effect, due to the nearly uniform stress distribution existing in the interior of the plate. For the small welded plate test specimens, F c = 2.19 for c = 0 and F c = l for c-----2.5 mm. For the large welded plate test specimens, Fc = 2.54 for c = 0 and Fc = 1 for c = 5 mm. All the results of the calculation of Fo for the welded plate test specimens investigated are given in Ref. 24, and the results from this calculation are used in the fatigue life predictions presented in the following.

3.1.2 Stress intensity factor due to welding residual stresses The stress intensity factor due to the welding residual stress field can be calculated from

C

/c r O'r,X, (25)

where o'rC0 is the welding residual stress distribution in the uncracked body at the line of potential crack growth.

Fatigue crack growth in residual stress fields may be analysed by using the crack closure model presented, if the following procedure is employed. The residual stress field is transformed into an equivalent, remote uniform stress, which results in the same stress intensity factor at the crack tip as the residual stress field. The method used to take into account the effect of the residual stresses is based on the concept that the primary effect of the residual stress field is to change the local mean stress. Thus, the effect on crack closure and hence on the fatigue crack growth of the change in the mean stress caused by the residual stress field is considered.

The stress intensity factor due to a remote, uniformly distributed stress, S~ a is given by

K = Fs . FE" FT" FG" Seq" ~f~. (26)

246 I. B. Ibsa, H. Agerskov

Equating the expressions in eqns (25) and (26) gives the equivalent stress of the residual stress field:

C

_ 2 I O'r(X)_..dx

For the welded plate test specimens analysed, the through thickness variations of the residual strains and stresses have been determined. 12,25 In Ref. 25, the neutron diffraction technique was used to measure the true bulk values rep- resentative of the bulk state of strains and stresses in the welded plate test specimens. On the basis of the residual strain measurements on the welded plate test specimens, discrete values of the residual stresses were estimated. The residual stress distributions through the thickness of the specimen are shown as open and solid symbols in Fig. 9 for the test specimen of thickness 16 mm (DTH16) and 8 mm (DTH08), respectively.

The estimated residual stress fields as shown in Fig. 9 are used in the calculation of the fatigue crack growth life of the welded plate test specimens in Sections 3.2-3.3.

30o ~ ~ 300

~ ~ DTHI6 zT- 250-~ ~, : : : : : DTH08 /~250

200 , ¢, 200

150 , / - 150

.100 1 O0 ,

~ 50 5 0 m

° 0 . . . . . . . . . . . . ~ . . . . . . , . . . . . . - / . . . . . . . . . . . . o

- 50 - i / ' - 5 0 ,~__~___;'

-1oo. 1 -ioo

I

-150 [ -150 0.0 0.'I 0.'2 0.~3 0.'4 0 .50 . f6 0.'70.r8 0.'9 1.0

x / t

Fig. 9. Estimated distribution of residual stresses, ~r acting in loading direction, as a function of relative position xlt in plate.


3.1.3 Parametric studies In order to investigate the possible scatter in the test data due to the variation in the crack: shape (c/b), and the initial crack size (cO, calculations were carded out using the crack growth computer program FAWS. In the calculations, the fatigue crack growth in the 8 mm thick test specimen (DTH08) was simulated for CA loading at R = - 1.

The crack growth coefficients C and m in eqn (2) were determined by a linear regression of crack growth rate data using the method of least squares. 12 The values of C and m found were compared to the results of Ref. 26, in which crack growth rate measurements were carried out for the same type of material. On the basis of this comparison, C and m were determined as

m = 3.1 (28 )

m C = 5.1- 10-12iMPax/-~)31. (29)

The values of m and C obtained in Ref. 26 and given in eqns (28)-(29) may be considered as being only minimally influenced by crack closure due to the high stress ratio applied, R = 0.6 and the plane strain condition, which may be assumed to exist in the test specimen used in Ref. 26.

For the material investigated, AKeff, th was estimated as

AKeff.t, = 2.5 MPa~j-m (30)

based on test results tabled in Ref. 22. The calculations were performed using the applied (full) stress ranges, that

is neglecting the influence of crack closure and assuming that the welding residual stresses are +fy all through the thickness of the specimen. The possible scatter in the fatigue life due to the variation in the initial crack size is shown in Fig. 10 for the fixed value of clb = 1/4. In the calculations, the final crack size was chosen as ce = 4 mm. This crack size corresponds to the final or critical crack size observed in the fatigue tests. The fixed value of the crack shape parameter, c/b = 1/4 in Fig. 10 was chosen based on experimental observations of the crack shape using the dye marking technique. ~2 The initial crack size was varied between ci = 0.075 mm and ci = 0.4 mm, which may be considered as a lower and an upper bound, respectively, for the initial crack size in the welded plate test specimens investigated, cf. Ref. 27. As may be seen in Fig. 10, the initial crack size has some influence on the calculated fatigue lives. Especially for low applied stress ranges, the initial crack size becomes a dominant factor as can be seen in Fig. 10. For comparison, the mean SN-


1000 I

s 5-

lOOg

N e

~'J E i

IO 10

I I I I I I I I I I I I I I

I I . . . . . . . e t : 0 . 0 7 5 Intla

¢11 J tJJ IIIII I l l t l I f l f l ,llll

c / b = 1 / 4

I I I IIII i 0 5

I I I | 1 1 I I I I I I l l I I 1 ' ' " I I I M e l t l l S - N l i l l e o f 1 i I t e n t d a t a ( D T E I 0 8 ) [ I [ i

I I Letff" I I I I l l J '~~. IJl[I

IIIIll I I I I I I I I I I I 1 I I I I I I

I Jlfll IIlll

I 0 ~

~ -7~'t-.1-11 I Iq" t l I l l l l IIIII IlllJ

rl I 0

Number of c y c l e s

Fig. 10. Influence of initial crack size on calculated fatigue lives for the welded plate test specimen, DTH08 .

line from the test series on the 8 mm thick specimens under CA loading at R = - 1 is also shown in Fig. 15. 20 As may be seen in Fig. 10, there are considerable differences between the calculated fatigue lives and the mean fatigue lives of the test data. The main reasons for this discrepancy are--to the authors' opinion--that the effect of crack closure is not included in the calculations shown in Fig. 10, and that the residual stresses are overestimated by using a value equal to +fy throughout. In Fig. 11 is shown the possible scatter

r ~

m

100001 B 7 e I

5

4

$

lOOg

o fi

4

10 1 0 '

i i i i l l t I I I l l

I[ I I I l l l I I I l l l

I I I l l l l

e . l b = ;_.'~ i~d-,91b e " ~

!! I I I I

II II II

e~ = 0.1 m m

I

I , !

I I

II IIIIIJ . . . . . . . . 10 5

N u m b e r o f c y c l e s

I l l l l I I I : : i i i i I l r t l i i I I I I I I I

i ! !Mean S-N line of~4~ [[ test data (DTII08I[-I-~ 1 1 , ~ / i , i l l l

i Ill

I ~", , , ,ET'I '~I.J I I I

I Jill I I [ l l t l l l l

} IlJll

II IIIIJ , i i l l I

10 o 10 7

Fig. 11. Influence of crack shape on calculated fatigue lives for the welded plate test specimen, DTH08 .


in the fatigue life due to the variation in the crack shape parameter, c/b for the fixed value of ci = 0.1 mm and cf = 4 mm. As may be seen in Fig. 11, the crack shape parameter, c/b has a considerable influence on the calculated fatigue lives. The two extreme cases in Fig. 11 are c/b = l, which corresponds to a half circular surface crack, and c/b = 0, which corresponds to a tunnel crack.

In the final calculations of the fatigue life of the welded plate test specimens given in Sections 3.2 and 3.3, an initial crack size of ci = 0.1 mm and a crack shape parameter of c/b -- 1/4 were used.

3.2 Constant ampl i tude loading

For the welded plate test specimens of thickness 8 mm (DTH08), constant amplitude load test series at two different applied stress ratios, R - - 0 and R = - 1 , have been carried through. For the test specimens of thickness t = 16 mm (DTH16), one constant amplitude test series at R = - 1 was carded through. "['he crack growth calculations are compared with the test results of these three CA test series. The calculations have been performed using the computer program FAWS.

In Figs 12 and 13 are shown results of crack growth calculations for the

C o n s t a n t amplitude l o a d i n g , R = - 1 .

S m a l l t e s t s p e c i m e n s (DTH08).

I ]

I I

(1)/ l

l O 0 0 t

i' 6

?-

~ 3

~ 2

I00

71

10 '

Il l l

, , i , , , g ,

10 5

I l l l I I I I I I I I l l I l l l l I I I I I IIIII JI

I I l I I I I I l l . " ~ " ~ , J ]1111 I I I I I I l l l l l I I

' ~ ' 6 ' , .~ , ,

10 6

II

II I II

III

! !ll 10 7

N u m b e r o f c y c l e s , N

Fig. 12. Crack growth calculations for DTH08 test specimen under CA loading at R = - 1 , considering the effect of welding residual stresses. (1) Full stress ranges. (2) Actual welding

residual stresses applied. (3) Mean SN-line of fatigue test series.


~D

1000 s

100 9 7 6

Constant amplitude loading, R = -i.

Large test specimens (DTIII6).

I II i II I i l

';"J J

I I i l l i

i J l i I I

i ....

J i I ] l i I I ' ~ i I I I

[ ' T ' ~ . . . . l i l l l 5 r ~ , ,~ , ~ , ~ , ~

lO ' 1 0 ~ 1 0 6 . tO 7

N u m b e r of cycles, N

Fig. 13. Crack growth calculations for DTHI6 test specimen under CA loading at R = - 1 , considering the effect of welding residual stresses. (1) Full stress ranges. (2) Actual welding

residual stresses applied. (3) Mean SN-line of fatigue test series.

Constant amplitude loading, R = -I. • .... Test results

Small test specimens (DTHO8). _ _ Prediction

'°°°'I IIJl F rl "f i~ll i Jl 6

~ l i I J ' " C d

l o o ,,,,, I Nt-~--~ r a i l l i l i i l l i I ?

10 ' 10 5 10 6 10 7


Fig. 14. Comparison of experimental and predicted fatigue lives for DTH08 test specimens under CA loading at R = - 1 .


welded plate test specimens considering the effect of the residual stresses. In the calculations, it is assumed that there is no plastic deformation remaining ahead of and behind the physical crack tip. Thus, Kop = 0 is assumed for all applied stress intensity factor ranges. For comparison, the mean SN-lines from the fatigue test series under CA loading at R = - 1 are plotted in Figs 12 and 13. These figures also show the SN-lines resulting from crack growth calculations using (full) applied stress ranges, i.e. assuming the residual stresses to be equal to +fy.

As may be seen in Figs 12 and 13, the residual stresses have a considerable influence on the calculated fatigue life for the high applied stress ranges, especially for the small test specimens. However, for the low applied stress ranges approaching the threshold level or fatigue limit, the effect of the residual stresses diminishes, and the calculated curves converge at the threshold level. Figures 12 and 13 also show that there are considerable differences between the calculated results and the mean SN-line for the fatigue test series, when crack closure is not considered in the calculations. The largest differences exist for the calculation using the (full) applied stress ranges.

Figures 14-16 also show the prediction results and the fatigue test results for the constant amplitude test series. However, for the predictions shown in Figs 14-16, both the influence of crack closure and of the actual welding residual stresses on the fatigue crack growth are considered. In these and in the following predictions, Irwin's plane strain constraint factor, ot = 1.73 is used. For the prediction shown in Fig. 16 (CA, R = 0), the crack shape parameter, c/b was chosen to 1/3 based on the experimental observations of the crack shape using the dye marking technique. 12

As may be seen in Figs 14-16, there is very good agreement between the prediction results and the fatigue test results for the welded plate test specimens investigated. Furthermore, it may be seen from Figs 14 and 16 that both the test results and the prediction for the small test specimens show signifi- cantly shorter fatigue lives at R = 0 than at R = - 1 , especially for the high applied stress ranges. The main reason for these differences in fatigue life is that the effective stress ranges and stress intensity factor ranges are larger for the case of R = 0 than for R = - 1, due to the relatively low tensile residual stresses present in the small test specimens.

3.3 Stochastic loading

3.3.1 Description of load spectra In the experimental investigation carded out at the Department of Structural Engineering of the Technical University of Denmark, three different load spectra have been applied. 2°,21 These spectra are denoted BROAD64, PMMOD64 and NARROW64. The spectrum BROAD64 was evaluated from a truncated


,o

Constant amplitude loading, R = -I.

Large test specimens (DTHI6).

1000 o 8 ? 6

,I

8

2

lOOg 8 !!-

2 1 0 '

Ill[ fill IIII fill

IIII Iltl

I1: [ I I I I l l [

I111

I I l l I I l l II

,.~ Illl

; 1 1 1 I I I I I I I I I l l l Il t l

2 , e~O e

N u m b e r of c y c l e s , N

• - ° , * T e s t r e s u l t s

_ _ P r e d i c t i o n

I I I I I J

llIfli iJl l[I Irlr I IIIrll I

"..11 I I I I F 'M II I

[trill

Fig. 15. Comparison of experimental and predicted fatigue lives for DTH16 test specimens under CA loading at R = - 1 .

v

6

O~

1000

100

10"

Constant ampl i tude loading, R = 0.

Small t es t spec imens (DTH08).

II II

IlJI

I I I

I

o o o o o T e s t r e s u l t s

rill ir l I ~

I I I I I I IIJ I I I I I Ill l l I Ill i,l~, J ,,, I I I ~'~" ~ . . . . . ~ . . . . . . ~,~,

10 5 i 0 e


__ Prediction

I l l l l I I I I I I I I I J l [ l l I I IIIII

IflIJ lllll I

flJ" II ~ - ' l

"+44-- - - - .A-- I I I

II ~,1 4 10 7

Fig. 16. Comparison o f experimental and predicted fatigue l ives for D T H 0 8 test specimens under C A loading at R = 0.


white noise spectral density function. The PMMOD64 spectrum was evaluated from a modified Pierson-Moscowitz wave elevation spectrum. The narrow band spectrum NARROW64, was evaluated from a band limited block spectrum.

The main characteristics of the three spectra used are given in Table 1. The Root Mean Square (RMS)- -and Root Mean Cube (RMC)---values given in Table 1 correspond to a maximum load level equal to the number of load levels, i.e. 64. The irregularity factor, L is defined as the number of positive- going mean-value crossings divided by the number of maxima. All three spectra used represent stochastic processes, which are symmetric with respect to zero. Further details of the load simulation procedures and the load spectra used may be found in Refs 20 and 21.

3.3.2 Crack growth simulation In the experimental investigations, 2°,21 the stress parameter used to represent the fatigue test results of the variable amplitude test series is the equivalent constant amplitude stress range, AS~qu defined as

k r ~ - ~ - _ 1 l / inCA

where k = number of different stress range levels n; = number of cycles of stress range AS~ AS~ = variable amplitude stress range

N = total number of cycles = n,. i=1 "

mcA = slope of corresponding constant amplitude SN-line.

TABLE 1 Characteristics of Load Spectra Used

(31)

Spectrum BROAD64 PMMOD64 NARROW64

Number of load levels 64 64 64 Irregularity factor, I 0.745 0.817 0.987

Minimum load range 1 1 1 Maximum load range 63 63 63 RMS (rainflow count) 18.6 19.8 22.9 RMC (rainflow count) 21.5 22.6 25.2

254 J. B. Ibso, H. Agerskov

From the constant amplitude (CA) test series at R = - 1, the values of mcA were determined as mCA = 2.85 and mCA = 3.10 for the test series on the small test specimens (t -- 8 ram) and the large test specimens (t = 16 mm), respectively. The cycle counting method used in Refs 20 and 21 was the rainflow counting method.

For the crack growth calculations, the same sequence and magnitudes of the applied stresses are used as for the fatigue tests carried out. In the crack growth calculations for variable amplitude loading, the same values of the initial crack size, ci, semi-axis ratio, c/b, crack growth coefficients C and m, etc., as used in the crack growth calculations for CA loading at R = - 1 are applied. Thus, the influence of crack closure and the welding residual stresses on the fatigue life under both constant amplitude and stochastic loading are focused.

Comparisons between predictions and fatigue test results for the small plate test specimens (DTH08) for stochastic loading are shown in Figs 17-19. As can be seen in these figures, there is in general a good agreement between prediction and fatigue test results for all three spectra analysed. For the large plate specimens (DTH16), similar results are presented in Figs 20-22. As may be seen in Figs 20-22, there is very good agreement between prediction and fatigue test results for the high applied stress ranges. However, for the low applied stress ranges, the test results show shorter fatigue lives than the prediction. The exact reason for the deviations at the low applied stress ranges has not been clarified. However, possible differences between the crack shape under CA loading and stochastic loading may explain the results at the low applied stress ranges. Thus, in Ref. 28 it was experimentally observed that VA tests gave more initiated cracks than CA tests at the low applied stress ranges, resulting in a higher degree of crack coalescence. The lower crack semi-axis ratios for VA loading will lead to higher crack growth rates as compared to CA loading.

In Tables 2 and 3 are given the predicted values of NvA/NcA for different equivalent constant amplitude stress range levels. NCA is the number of cycles to failure at constant amplitude loading at R = - 1, and NVA is the number of cycles to failure at variable amplitude loading including all stress cycles (i.e. no truncation), at the same stress level. For comparison, the corresponding ratios of NvA/NcA determined from the linear regression lines of the fatigue test series are also given in Tables 2 and 3. 2°'21 As can be seen in these tables, the predicted values of NvA/NcA are less than 1 in all cases. This is also the case for the experimental values of NvA/NcA except for one case.

The predicted results given in Tables 2 and 3 show that the value of NvA/NcA increases with decreasing values of the equivalent constant amplitude stress range. At the higher equivalent stress ranges, the higher magnitudes of the maximum applied tensile stresses and in particular the higher magnitudes


<1

O9

Q)

10009 6 7

0

6

4

1 0 0 6 0 7

8

5

1 0 4

Stochastic loading, NARROW64 spectrum. Small tesL specimens (DTH08).

I111 1111

IIII I

4 8 8 1 0 5

I1111 I1111 Ilil[

4 6 ~ 0 6

• • • • • Test results _ _ Prediction

Ill

:. 1]11 4 6 ~ 2 4

1 0 ~


Fig. 17. Comparison of prediction and fatigue test results for small test specimens with NAR- ROW64 spectrum.

1000

O~

M I00

S t o c h a s t i c l o a d i n g , P M M O D 6 4 s p e c t r u m . Small test specimens ( D T H 0 8 ) .

i l I I I I t l

II

III 1

I I I I

: illl ,'Ill L I ! l l tl II

2 ~'''~ . . . . .

10 ' 1 0 5 1 0 6

N u m b e r of cycles , N

• .... Test results _ _ Prediction

I Illl

I I I I I •

1 10 7

Fig. 18. Comparison of prediction and fatigue test results for small test specimens with PMMOD64 spectrum.


f/3 " 0

a)

09

Stochastic loading, BROAD64 spectrum. Small test specimens (DTH08).

lo0o Illll I ,'"',,, ' IIIII i 1 1 1 1 . I I I I I i i i i i !!!!! ! . . . . .

IIIII " ~ , ~ .I III

iiiii i " ~

lO0 !11 I I I I I : i , I I I . . . . . .

!! 7 ' , i i i i i

,~ , ~, . , , ; , ~

10 ' 10 5 10 ~ N u m b e r of c y c l e s , N

• • ' ' " Test results _ _ Prediction

• I

Lli !

1 0 7

Fig. 19. Comparison of prediction and fatigue test results for small test specimens with BROAD64 spectrum.

1000~ 6 7

4

<1

r~

u~ 1 0 0

6

5

10 4

Stochastic loading, NARR0W64 spectrum. Large test specimens (DTH16).

JrrI'

i j i IIII

lli i I L.

I 4 6 ~.05 2

FI IIIII

4 6 ~0 ~

• • • = * T e s t r e s u l t s _ _ P r e d i c t i o n

[i Il l ]

I

i rJ . . II ~ . 1 I I I "h'l', II

• . 10 7 2 4


Fig. 20. Comparison of prediction and fatigue test results for large test specimens with NAR- ROW64 spectrum.


1000

v

o

100

1 0 '

Stochastic loading, PMMOD64 . spectrum. Large test specimens (DTHI6).

Illlt [ l l i i i l l l i

i l

I I I I I I I I I I I

I 1 1 1 1

II 2 4 6 8 2

10 5

I I I I IL Illl llli L

Illl

4 @ 8 10 6


Illll IIIII IIIII IIIII IIItl

"- [ i l l i i l i l l i i l i i

II l l l illll

~ 4 6 8 10 ~

2 4


Fig. 21. Comparison of prediction and fatigue test results for large test specimens with PMMOD64 spectrum.

1000

~oo i

e :

1 0 '

Stochastic loading, BROAD64 spectrum. Large test specimens (DT}tl6).

I l l l l

I l l l l

L ilIIL

Lql l i l l i L i l l i i ] l i l i i i lL t l l l l

4 6 8 2 10 ~

I l l l

Illl

liii [BIB

I I I 1 "~

lill i 1 1 1

I l l l 4 6 8 2

10"


lllll

tl

L hi lllli IIIII

II 4 6 8 2 4

10 7


Fig. 22. Comparison of prediction and fatigue test results for large test specimens with BROAD64 spectrum.


TABLE 2 Values of NvA/NcA at Different Stress Range Levels, ASequ for Small Test Specimens (DTH08)

ASeq u = 120 M P a Aaeq u = 200 MPa ASeq, = 250 MPa

NvA/NcA NvA/NcA NvA/NcA NvA/NcA NvA/NcA NvA/NcA Spectrum Predicted Test Predicted Test Predicted Test

NARROW64 0.95 0.84 0.76 0.82 0.65 0.81 PMMOD64 0.88 1.03 0.47 0.63 0.41 0.50 BROAD64 0.75 0.49 0.38 0.44 0.34 0.42

TABLE 3 Values of NvA/NcA at Different Stress Range Levels, ASoqu for Large Test Specimens (DTH16)

ASeq u ~- 150 MPa AS~q, = 200 M P a ASeq u = 250 MPa

NvA/NcA NvA/NcA Nva/Nca NvA/NcA NvA/NcA NVA/]VCA Spectrum Predicted Test Predicted Test Predicted Test

NARROW64 0.93 0.58 0.81 0.62 0.69 0.66 PMMOD64 0.83 0.50 0.63 0.58 0.49 0.66 BROAD64 0.74 0.39 0.55 0.41 0.42 0.42

of the applied compressive stresses for the variable amplitude load cases play a dominant role, and this results in a significant acceleration of the crack growth as compared to constant amplitude loading. In particular, at small crack depths, where the plastic stretches in the wake of the crack are being built up, the higher magnitudes of both the tensile and compressive stresses for the spectra analysed result in an acceleration of the crack growth as compared to CA loading.

Different studies in the literature--both experimental and theoretiqal--have also shown that crack behaviour, and in particular the short crack behaviour depends not only on AK, but also on the applied load level, especially at low applied stress ratios, see e.g, Refs. 4, 29 and 30.

At the lower equivalent stress ranges, a larger part of the applied stress ranges becomes effective and contributes to the crack growth. This particularly applies for the stress ranges under CA loading. This effect is due to the tensile residual stresses present in the welded plate test specimens. At the lower applied equivalent stress ranges, the predicted load interaction effects are therefore less pronounced for the welded plate test specimens analysed.


4 CONCLUSIONS

The analyses of the present study have shown that crack closure mechanisms are of paramount importance for the physical understanding and explanation of fatigue crack growth under various load conditions and under different stress states. Fatigue crack growth phenomena such as delayed retardation following a single tensile overload, crack growth acceleration following a compressive underload and crack growth retardation following multiple tensile overloads may be explained on the basis of crack closure analyses.

The parametric studies of the crack growth behaviour of surface cracks at welded joints show that the following factors are of great importance, and thus in general they all ought to be included in fatigue life predictions of welded joints:

- - welding residual stresses - - crack closure

stress concentrations - - crack shape.

It is possible to include stress concentrations and welding residual stresses in the fracture mechanics analysis. Methods to include in the analysis of the influence of stress concentrations and welding residual stresses upon crack closure and fatigue crack growth have been developed in the present study.

Using a crack closure model as presented herein and including the influence of stress concentrations and welding residual stresses results in good agreement between predictions and fatigue test results under CA loading at R = 0 and R = - 1 for the welded plate test specimens analysed. On the other hand, using full stress ranges in the crack growth calculations, i.e. neglecting the influence of crack closure and assuming the residual stresses to be equal to the tensile yield stress, results in a significant underestimation of the fatigue life under CA loading at R = - 1 for the plate test specimens.

The analytical results for stochastic loading show that acceleration effects dominate the fatigue crack growth for the load spectra analysed. This main result is in good agreement with the findings of the experimental investigations. The main reason for the differences in the damaging effect between the load spectra analysed, and between the load spectra and CA loading at R = - 1, respectively, is the differences in the magnitudes of the applied stresses, especially the differences in the magnitudes of the applied compressive stresses. In particular at small crack depths, where the plastic stretches in the wake of the crack are being built up, the higher magnitudes of both the tensile and the compressive stresses for the load spectra analysed result in an acceleration of the crack growth as compared to constant amplitude loading at R = - I .


Both the analytical results obtained by use of a fracture mechanics analysis, where crack closure and the actual residual stresses are included, and the experimental investigations in which a large number of fatigue tests with stochastic loading has been carded out, show that quite unconservative results may be obtained by use of the Miner rule for the types of loading investigated.

REFERENCES

1. Elber, W., The significance of fatigue crack closure. ASTM STP 486 (1971) 230-242.

2. Paris, P. C. & Erdogan, F., A critical analysis of crack propagation laws. J. Basic Engng, Trans. ASME D85 (1963) 528-534.

3. Paris, P. C., Gomez, M. P. & Anderson, W. E., A rational analytic theory of fatigue. The Trend in Engng, 13 (1961) 9-14.

4. Newman, J. C., Jr., A nonlinear fracture mechanics approach to the growth of small cracks. AGARD, CP-328 (1983).

5. Wang, G. S. & Blom, A. F., A strip model for fatigue crack growth predictions under general load conditions. Engng Fracture Mechanics, 40 (1991) 507-533.

6. Dugdale, D. S., Yielding of steel sheets containing slits. J. Mech. Phys. Solids, 8 (1960) 100-104.

7. Barenblatt, G. I., Adv. Appl. Mechanics, 7 (1962) 55-129. 8. Newman, J. C., Jr., A crack closure model for predicting fatigue crack growth

under aircraft spectrum loading. ASTM STP, 748 (1981) 53-84. 9. Dill, H. D. & Saff, C. R., Spectrum crack growth prediction method based on

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10. Fuehring, H. & Seeger, T., Structural memory of cracked components under irregular loading. Fracture Mechanics, ASTM STP, 677 (1979) 144-167.

11. Tada, M., Pads, P. C. & Irwin, G. R., The Stress Analysis of Cracks Handbook. Del Research Corporation, MO, 1973.

12. Ibs~, J. B., Fatigue life prediction of welded joints based on fracture mechanics and crack closure. Ph.D. thesis, Report No. R 322, Department of Structural Engineering, Technical University of Denmark, Lyngby, Denmark, 1995.

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15. Willenborg, J., Engle, R. M. & Wood, H., A crack growth retardation model using an effective stress intensity concept. AFFDL-TM-FBR-71-1, Air Force Flight Dynamics Laboratory, Wright Patterson Air Force Base, OH, USA, 1971.

16. Nowack, H., Trautmann, K. H., Schulte, K. & Ltitjering, G., Sequence effects on fatigue crack propagation; mechanical and microstructural contributions. Frac- ture Mechanics, ASTM STP, 677 (1979) 623-628.

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18. Irwin, G. R., Plastic zone near a crack and fracture toughness. Proc. 7th Saga- more Conf., 1960, pp. IV, 63-77.


19. Jacoby, G. H., Nowack, H. & van Liepzig, H. T. M., Experimental results and a hypothesis for fatigue crack growth under variable-amplitude loading. Fatigue Crack Propagation Under Spectrum Loads, ASTM STP, 595 (1976) 172-183.

20. Ibs¢, J. B. & Agerskov, H., Fatigue Life of Offshore Steel Structures under Stoch- astic ~gading. Dept of Structural Engineering, Technical University of Denmark, Lyngby, Denmark, Report No. R 299, 1992.

21. Pedersen, N. T. & Agerskov, H., Fatigue Life Prediction of Offshore Steel Struc- tures under Stochastic Loading. Dept of Structural Engineering, Technical Uni- versity of Denmark, Lyngby, Denmark, Report No. R 274, 1991.

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24. Yamada, K. & Agerskov, H., Fatigue Life Prediction of Welded Joints Using Fracture Mechanics. Dept of Structural Engineering, Technical University of Denmark, Lyngby, Denmark, Report No. R 255, 1990.

25. Lorentzen, T. & Ibs¢, J. B., Neutron diffraction measurements of residual strains in offshore welds. Materials Sci. Engng, A197 (1995) 209-214.

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27. Yamada, K. & Nagatsu, S., Evaluation of scatter in fatigue life of welded details using fracture mechanics. Struct. Engng/Earthquake Engng, JSCE 6 (1989) 13-21

28. Slind, T., Long life testing of welded 370 MPa YS steel specimens in air and in seawal:er with cathodic protection. Proc. Nordic Conf. Fatigue, ed. A. F. Blom, EMAS Publishers, UK, 1993, pp. 91-105.

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An Analytical Model for Fatigue Life Prediction Based on Fracture Mechanics and Crack Closure (1996)

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