Mixed-Mode Fatigue Crack Propagation in Thin

8/13/2019 Mixed-Mode Fatigue Crack Propagation in Thin

1/20

http://sdj.sagepub.com/Design

The Journal of Strain Analysis for Engineering

http://sdj.sagepub.com/content/38/6/557Theonline version of this article can be found at:

DOI: 10.1243/030932403770735917

2003 38: 557The Journal of Strain Analysis for Engineering DesignA. R Gowhari-Anaraki, S J Hardy and R Adibi-Asl

Mixed-mode fatigue crack propagation in thin T-sections under plane stress

Published by:

http://www.sagepublications.com

On behalf of:

Institution of Mechanical Engineers

can be found at:The Journal of Strain Analysis for Engineering DesignAdditional services and information for

http://sdj.sagepub.com/cgi/alertsEmail Alerts:

http://sdj.sagepub.com/subscriptionsSubscriptions:

http://www.sagepub.com/journalsReprints.navReprints:

http://www.sagepub.com/journalsPermissions.navPermissions:

http://sdj.sagepub.com/content/38/6/557.refs.htmlCitations:

by guest on May 21, 2011sdj.sagepub.comDownloaded from
http://sdj.sagepub.com/http://sdj.sagepub.com/http://sdj.sagepub.com/http://sdj.sagepub.com/content/38/6/557http://sdj.sagepub.com/content/38/6/557http://www.sagepublications.com/http://www.imeche.org/homehttp://sdj.sagepub.com/cgi/alertshttp://sdj.sagepub.com/subscriptionshttp://sdj.sagepub.com/subscriptionshttp://www.sagepub.com/journalsReprints.navhttp://www.sagepub.com/journalsReprints.navhttp://www.sagepub.com/journalsPermissions.navhttp://sdj.sagepub.com/content/38/6/557.refs.htmlhttp://sdj.sagepub.com/content/38/6/557.refs.htmlhttp://sdj.sagepub.com/http://sdj.sagepub.com/http://sdj.sagepub.com/http://sdj.sagepub.com/content/38/6/557.refs.htmlhttp://www.sagepub.com/journalsPermissions.navhttp://www.sagepub.com/journalsReprints.navhttp://sdj.sagepub.com/subscriptionshttp://sdj.sagepub.com/cgi/alertshttp://www.imeche.org/homehttp://www.sagepublications.com/http://sdj.sagepub.com/content/38/6/557http://sdj.sagepub.com/


2/20

Mixed-mode fatigue crack propagation in thinT-sections under plane stress

A R Gowhari-Anaraki1

, S J Hardy2* and R Adibi-Asl

3

1Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran2School of Engineering, University of Wales, Swansea, UK3Department of Mechanical Engineering, University of Tehran, Iran

Abstract: Data that can be used in the fatigue analysis of at T-section bars subjected to axial

loading and local restraint are presented. The paper describes how nite element analysis has been

used to obtain stress elds in the vicinity of a crack or crack-like aw introduced into the llet (i.e.

high-stress) region of the component. The effect of both the position and inclination of the crack has

been investigated. The inclination of the crack to the transverse direction is varied in such a way that a

combination of mode I (tension opening) and mode II (in-plane shear) crack tip conditions are

created in the component when subjected to axial loading which is applied to the entire at shoulder

of the projection. Linear elastic fracture mechanics nite element analyses have been performed, and

the results are presented in the form ofJ integrals and notch and crack conguration factors for a

wide range of component and crack geometric parameters. These parameters are chosen to be

representative of typical practical situations a nd have been determined from evidence presented in the

open literature. The extensive range of notch and crack conguration factors obtained from the

analyses are then used to obtain equivalent prediction equations using a statistical multiple non-linear

regression model. The accuracy of this model is measured using a multiple coefcient of

determination, R2, where 0< R2


3/20

Ni initiation fatigue life

Np propagation fatigue life

Nt total fatigue life

P uniform pressure

Q vectors

r llet (notch) radius

r, y polar coordinates associated with the

crack tip

R2 multiple coefcient of determination

t component thicknessTk traction vector

w projection width

W strain energy density

x, y crack coordinate system

x0, y0 global coordinate system

a crack angle

b material constant

Du,Dv relative displacement of points along

opposite sides of the crack face

De strain range at the notch

De0 nominal (axial) strain range

Ds stress range at the notch

Ds0 nominal stress range

e0f fatigue ductility coefcientyc crack growth direction

n Poissons ratio

sx,y directional stresses

s0 nominal (axial) stress

s0f

fatigue strength coefcient

txy shear stress

j position around the llet

1 INTRODUCTION

Projections on plates, bars and tubes are often used as a

means of transmitting axial load between two compo-

nents, e.g. T-shaped at bars, shouldered plates/shafts/

tubes, wide grooves, lleted transitions and many other

geometric shapes with similar stress concentration

features (see examples in Fig. 1). For remote loading

conditions, the point of application of the load is usually

far removed from the region of stress concentration (see

Fig. 2a) and does not inuence that local stress eld.

However, when the load or the reaction of the loading is

applied at or near to the region of high stress gradient,

such as for these `loaded projections, the stress intensity

factors can be signicantly higher than for the equiva-

lent remote loading case (see Fig. 2b). This is because

there is both a tensile component and a bending

component of load at the projection, as illustrated in

Fig. 2c.

Fatigue failure is a major consideration in mechanical

design. Nearly all fatigue failures initiate at stress

concentration features, i.e. at the point of highest stress

under cyclic or repeated loading conditions. In general,

the initial crack (or any crack-like aw, void, defect,

etc.) will develop in three stages, as presented in

reference [1], namely initiation, propagation and frac-

ture. Crack initiation is analysed at the microscopic

level, while for crack propagation the continuum

mechanics approach based on a macroscopic scale is

used. Paris [2] showed that the stress intensity factor and

3B2 Version Number 7.51a/W (May 2 2001) j:/Jobsin/M11069/S03603.3d Date: 28/10/03 Time 09:14am Page 558 of 576

Fig. 1 Typical stepped at bar geometries

Fig. 2 Typical example of a loaded projection and remoteloading

A R GOWHARI-ANARAKI, S J HARDY AND R ADIBI-ASL558

J. Strain Analysis Vol. 38 No. 6 S03603 # IMechE 2003

http://sdj.sagepub.com/http://sdj.sagepub.com/http://sdj.sagepub.com/http://sdj.sagepub.com/


4/20

J-integral are very important controlling factors in the

fatigue crack growth of cracked components.

In order to predict the strength of the cracked

component, the crack growth rate and the critical crack

size, an accurate value for stress intensity factor along

the crack front has to be known. Many experimental,

numerical and analytical solutions for the three basic

stress intensity factors Ki (i I, II, III) have beendeduced for varying crack sizes for relatively simplistic

structures with basic loading conditions. Paris and Sih[3] present a comprehensive handbook of such results. In

spite of wide-ranging applications of loaded projections,

relatively very little information on the resulting stress

intensity factors is available and, for the more realistic

complex shapes encountered in practice, further general

data are still required. Part of the reason for this lack of

systematic information is due to difculties involved in

obtaining analytical solutions. Whereas the stress

intensity factor in a remotely loaded cracked component

is controlled by the notch radius, r, projection length, t,

crack length, a, and the remotely applied load, P, as

shown in Fig. 2a, for cracked components with loaded

projections the projection length, h, and the combined

effect of tensile and bending loading also affect the stress

intensity factor, as illustrated in Figs 2b and c. For the

equivalent remote loading case, however, crack tip stress

elds have previously been studied, and data on elastic

stress intensity factors are readily available (e.g.

reference [4]). Also, the crack tip stress solution for

anked notches can be found in reference [5], for

example.

In the present paper, nite element analysis is used to

predict stress intensity factors for an external inclined

crack in a stepped at plate under complex projection-

loaded conditions, as shown in Fig. 3. This geometry

contains a stress concentration feature in the llet,

exacerbated by the projection loading, which is an

obvious source of crack initiation and propagation

under fatigue loading conditions. Modes I and II and

mixed-mode conditions are created by varying the

inclination of the crack, a, in the range 0908, and

stress intensity factors and J-integral values are pre-

sented for these modes under linear elastic conditions.

The results of the parametric survey are then used to

develop predictive equations that enable the notch and

crack conguration factors to be determined for therange of geometric parameters considered in this study.

These results are then used to obtain predictive

equations for stress intensity factor and J-integral values

that are based on the elastic stress concentration factor

for the geometry being considered. Thus, a direct link is

provided between the notch parameter (which can be

easily determined either from the predictive equations

presented here or via elastic nite element analysis) and

crack parameters, which normally require the use of

complicated linear elastic fracture mechanics analysis. A

methodology for determining fatigue life of components

with loaded projections is described that is based on

established theory coupled with these predictive equa-

tions.

2 GEOMETRY, LOADING AND BOUNDARY

CONDITIONS

Four dimensions are used to dene the geometry, as

shown in Fig. 3. They are the projection length, h, the

projection width, w, the plate width, d, and the llet

radius, r. Three non-dimensional parameters a re formed

by normalizing with respect to the plate width, d, i.e.

h=d, w=d and r=d.


Fig. 3 Component geometry and loading

MIXED-MODE FATIGUE CRACK PROPAGATION IN THIN T-SECTIONS UNDER PLANE STRESS 559

S03603 # IMechE 2003 J. Strain Analysis Vol. 38 No. 6



5/20

The range of dimensions selected for the parametric

study is consistent with the geometric cases covered by

Engineering Sciences Data Unit (ESDU) data item

69020 [6], and is considered to present a range of

practical interest. The selected ranges are

35h

d50:5

35w

d

51:5

0:25r

d50:05

The plate length, L, is made long enough to ensure that

uniform stress conditions are achieved away from the

llet. A plane stress assumption (i.e. thin plate) is made

for all geometries considered.

The loading condition consists of a remotely applied

axial stress,s0, reacted by a uniform pressure, P, across

the entire at section of the shoulder as shown in Fig. 3.

The at section of the shoulder has a width, b, where

b 12w d r 1and the nominal stress in the plate is given by

s0 2Pbd

P wd 2r

d 1

2

An edge through-crack or crack-like external aw is

assumed to emanate from the high-stress region around

the llet. It was found [1] that the maximum total mixed-

mode stress intensity factor, as well as the maximum J-

integral value, corresponded to a crack emanating from

a point around the llet located at j

25 from the

plate/llet intersection (see Fig. 3). The inclination of the

crack to the transverse direction, a, is selected in such a

way that a combination of mode I and mode II crack tip

conditions is created, i.e. 04a4 90. A range of cracklengths, a, is considered on the basis of the observations

of surface cracks in fractured standard NDT (non-

destructive testing) test specimens [7], i.e. a 0.52.5 mm. Many engineering components contain, or are

assumed to contain, such cracks for life assessment

purposes at the design stage. Such cracks may grow

owing to fatigue, corrosion, creep, etc.

3 FINITE ELEMENT ANALYSIS OF THE

CRACKED COMPONENTS

Finite element predictions have been obtained using the

standard linear elastic fracture mechanics facilities

within the ANSYS suite of programs [8]. Six- and

eight-noded, reduced integration, plane stress, triangu-

lar and quadrilateral elements were used with the crack

tip singularity represented by moving nodes to the

quarter-point positions [8]. A typical nite element meshis shown in Fig. 4. In preliminary work, solutions were

obtained with and without crack tip elements. The

results indicated that large discrepancies (*15 per cent)

in the predicted stress intensity factors were obtained if

crack tip elements were not used. The results presented

in this paper were therefore obtained using crack tip

elements. The results of a preliminary nite element

analysis under remote loading conditions were com-

pared with results obtained by Gray et al. [9]. This

comparison conrmed that the level of mesh renement

and crack tip elements used in the current study would

provide accuracy to within +2 per cent. Values for

Youngs modulus and Poissons ratio of 209 GPa and

0.3 respectively have been used throughout the analysis.


Fig. 4 Typical nite element mesh





6/20

4 RESULTS

4.1 J-integral values based on numerically determined

stresses and displacements

Expressions for KI and KII can be obtained in terms of

the nodal stresses ahead of the crack tip (i.e. sy andtxy),

i.e.

KI limr!0

sy

2prp 3KII lim

r!0txy

2pr

p 4

Similarly, expressions for KI and KII in terms of the

relative displacement of points, on opposite faces of the

crack, along the line of the crack, Du, a nd perpendicular

to it, Dv, are

KI limr!0

E

8

2p

r

r Du

! 5

KII limr!0

E

8

2p

r

r Dv

! 6

These stress components and relative displacements are

dened with respect to the coordinate system attached

to the crack tip, as shown in Fig. 5. A typical plot ofKIand KII is shown in Fig. 6 for a typical geometry (i.e.

h/d 0.5, w/d 3, r/d 0.05, a 0 and crack lengtha 0.5 mm) with a crack position around the llet,j, of258 (for reasons discussed later in this section). KI and

KII values have been extrapolated to r

0 for each of

the stress

y

0) and displacement

y

+180

results,

rather than use predictions at the crack tip.

The elastic J-integral value is then calculated by

substituting KI and KII values into the following

equations for plane stress (i.e. thin plate) conditions [10]:

J K2I K2II

E 7

JE

p

K2I K2II

q Ktm 8

where Ktm is referred to here as the total mixed-mode

stress intensity factor.


Fig. 5 Stresses and coordinate systems in the vicinity of acrack

Fig. 6 Typical nite element results based on stress and displacement extrapolation forh=d 0:5, w=d3and r=d

0:05 with a

0, a

0:5 mm and P

100MPa





7/20

It is very important not to confuse Ktm with the

effective stress intensity factor, Keff, where, for this

component, the effective stress intensity factor can be

roughly dened as

Keff

K2I bK2IIq

9

where b is a material constant, which reects the

sensitivity of the material to mode II. Further work on

the calculation of the b parameter and the validation of

equation (9) is in progress and will be reported at a later

date.

The variations ofKI, KII and J-integral values [from

equation (7)] around the llet are shown in Fig. 7 for the

same typical geometry as Fig. 6. Although the crack is

always perpendicular to the loading, the results show

that both mode I and mode II contributions are present.

This is due to the shear effect caused by the applied

pressure loading in the llet region of the component.

For all the geometries considered, it was found that

the magnitude of the crack stress intensity factors varied

considerably with both j and a, and the maximum value

ofKI occurred for a crack emanating from the position

j&25 from the plate/projection intersection (seeFig. 3). It was also found that the maximum value of

KII corresponded to j&50. The maximum J-integral

value occurred for a crack originating from j 308, butthe value atj 258 is very close to the maximum. Also,the maximum meridional (i.e. surface) stress is found to

occur at an angle j&25 (see section 4.3.1). Therefore,all the remaining results in this paper are for an edge

angle crack (with crack angle 0 4a4 90) originatingfrom j 25.

4.2 J-integral values based on the virtual crack

extension method

It is recognized by the present authors that the method

of calculation described in section 4.1, based onextrapolation, is not necessarily very accurate and

that, to obtain reliable results, a highly rened mesh

and tests to verify the mesh are necessary. In order to

validate the predictions based on this method,J-integral

values have also been obtained using a numerical

procedure based on the virtual crack extension method

(VCEM). This method only requires rather coarse

meshes in order to obtain good results.

In this section, elastic J-integral values have been

derived by means of VCEM, as follows:

J G

Wdy Tk qukqx ds, k 1, 2 10The VCEM procedure incorporated in the ANSYS nite

element program [8] has been used to evaluate the J

integrals. This evaluation was supported by a macro

program, written by the authors, using ANSYS para-

metric design language (APDL). This program can

conveniently be used as part of a post-processing

program and uses stress and displacement data from a

linear elastic fracture mechanics analysis to calculate the


Fig. 7 Variation in KI, KII and J integral with position around the llet for h=d 0:5, w=d 3 and r=d0:05 with a

0, a

0:5mm and P

100MPa





8/20

J-integral values. This procedure was carried out forfour separate contours around the crack tip, which are

shown in Fig. 8 for a typical geometry. Contour 1 is

around the outside of the mesh, and contours 2 to 4

encircle the crack with progressively smaller enclosed

areas. All contour integrals generally showed good path

independence, as illustrated in Table 1 where the J

values are normalized with respect to the value for the

outermost contour 1. The only case that exceeds a 1 per

cent deviation is the innermost contour, contour 4.

Consequently, the values for contour 1 were used

throughout this study.

Table 2 shows a comparison ofJvalues obtained bymeans of the numerically determined stresses with those

calculated using VCEM, for a range ofw/dvalues. It is

seen that the agreement between the two approaches is

very good, and this suggests that the method described

in section 4.1 has benets since it does not require the

generation of a macro-program.

4.3 Notch and crack conguration factors

Notch and crack conguration factors are very impor-

tant tools for fatigue crack initiation and fatigue crack

propagation life predictions respectively. Information

on stress intensity factors for anked notched compo-nents under simple loading conditions, e.g. remote

loading, is readily available [11,12]. Such data, however,

are not available for loaded projections, and in this

section simple parametric equations are presented to

assist in the design life assessment of thin T-section

plates, which eliminates the need for complex calcula-

tions of notch and crack conguration factors.

4.3.1 Notch conguration factors

The notch conguration factor (e.g. elastic stress

concentration factor) is dened as a geometric correc-

tion factor, Fn, which is a function of three non-

dimensional parameters, h/d, w/dand r/d, as follows:

Fn f hd

,w

d,

r

d

Kt 11

The elastic stress concentration factors, Kt, have been

obtained by dividing the predicted maximum local

elastic stress in the llet region by the nominal stress,

s0, from equation (2). For all geometries considered, the

maximum stress was in the meridional direction andgenerally close to the intersection between the plate and

the llet radius, where the J integral had a maximum

value (i.e. j&25).Using these Kt values, useful prediction equations for

Fn have been derived, based on the denition in

equation (11) and using a statistical multiple non-linear

regression model [13], as follows:

Fn Kt

0:91h=d2:11 0:26w=d1:44 0:57r=d0:92 0:14

h=d

1:68

w=d

0:25

r=d

0:49

12The accuracy of this equation has been assessed using a

multiple coefcient of determination [13], R2, where

04R2 4 1. This coefcient was found to be approxi-

mately 0.98, which demonstrates the accuracy of model

t to the data. The values ofFn that are obtained from

equation (12) compare favourably with direct nite

element predictions for some typical geometries, as

shown in Fig. 9. It can also be seen that they are very

close (within 2 per cent) to experimental results from

reference [14].


Fig. 8 J-integral contour paths

Table 1 NormalizedJintegrals for typicalcontour paths

Contour 1 2 3 4NormalizedJ 1 0.998 0.992 0.974

Table 2 Comparison of J-integral valuesobtained from stresses and VCEMfor h/d 0.5, r/d 0.2, a 0.5mmand a 408

Jintegral (N/mm)

w/d From stresses VCEM

1.9 0.97 0.952.3 4.37 4.512.7 12.20 11.963.0 22.47 22.34





9/20

4.3.2 Crack conguration factors

The stress intensity factor KN (where N I, II andcorresponds to the mode I and mode II stress elds) is

usually written as

KN Fcspa

p 13

where Fc is the crack conguration factor, which

accounts for such things as proximity effects of

boundaries, orientation of the crack, shape of the crack

and the restraints on the structure containing the crack.

It is with the determination of this geometrical crack

conguration factor that fatigue crack growth analysis

methods are concerned in the three stages of failure,

namely initiation, propagation and fracture. The analy-

sis of fatigue crack growth from its initial dimensions to

its nal critical stages requires the accurate calculation

of the instantaneous crack conguration factor.

For the purpose of the initiation stage analysis of the

shouldered plate with an axially loaded symmetrical

projection, an initial angle crack (or crack-like external

aw) emanating from a critical point along the llet

radius (i.e. a point with a maximum J-integral value at

j 25 as shown in Fig. 3) was assumed. Theinclination of this initial crack to the transverse

direction, a, was varied in the range 0 to 908, in such a

way that mixed-mode KI and KII crack tip conditions

were created. This kind of initial crack or crack-like aw

can be created practically under strain cycle fatigue

conditions or be introduced during manufacturing and

assembly processes, or are inherent in the basic metal.

Unfortunately, estimating this initial crack size involves

non-destructive crack detection and sizing or proof

testing, which is a very complicated engineering problem

and beyond the scope of this paper. However, the

denition of an initial crack has been the subject of

much controversy. Unfortunately, no satisfactory solu-

tion to this problem exists. Fatigue cracks start with

dislocation movement on the rst load cycle and end

with fracture on the last. Crack initiation lies somewhere

between the two. For the purposes of strain cycle fatigue

analysis, crack initiation is dened as a crack in the

structure or component that is the same size as the

cracks observed in the strain cycle fatigue specimen.

Frequently, this is the specimen radius, which is of the

order of 2.5 mm [15]. Dowling [16] proposed that strain

cycle fatigue data should be presented in terms of the

number of cycles required to reach a crack of xed

length. He found that, for steels with fatigue lives below

the transition fatigue life, cracks 0.25 mm long were

formed at approximately 50 per cent of the life required

for specimen separation. For smooth specimens with

longer lives, where the bulk behaviour of the material is

primarily elastic, the rst crack is observed just prior to

specimen fracture. Therefore, for design purposes, crack

initiation is dened as the formation of cracks between

0.25 and 2.5 mm long. It should be noted that the cyclic

behaviour of small cracks (less than 0.25 mm) is

different to that of long cracks under equal stress

intensities [17]. As a result, the analysis described in this

study does not apply to these very small cracks. How-

ever, in this part of the present study, a reasonable initial

crack length, a 0.52.5 mm, was assumed throughoutthe analysis. These crack sizes, which are easily seen at

610 to 620 magnication, can be related to engineer-

ing dimensions and can represent a small microcrack.

Using the methodology described in section 4.1, mode

I and mode II stress intensity factors (KI,KII) have been

obtained from the nite element predictions for more


Fig. 9 Variation in notch conguration factor with w/df or h=d0:5 and r=d 0:05





10/20

than 2000 combinations of crack length (a 0.52.5mm), crack angle (a 0908) and different compo-nent geometriesh=d, w=d, r=d, as dened in section 2).As an example, the variation in KI and KII with crack

angle is presented in Figs 10 to 12 for a range of

geometric parameters and a crack length of 0.5 mm. In

all cases, j 258.The lled-in symbols in Figs 10 to 12 are for the same

geometric parameters and enable direct comparison

between the three gures. A number of signicant

features can be identied:

1. For all geometries considered in this study, the mode

I stress intensity factor becomes equal to the mode II

stress intensity factor (i.e. KI KII) when the crackangle is 458. According to equation (7), this means

that the Jintegral also reaches a maximum when the

crack angle a 45.


Fig. 10 Variation in KI and KII with crack angle for h=d 0:5, w=d 1:5 and 3 and r=d 0:05 with a0:5 mm and P 100MPa

Fig. 11 Variation in KI and KII with crack angle for h=d 0:5 and 3, w=d 3 and r=d 0:05 with a0:5 mm and P

100MPa





11/20

2. The crack angle a 45 was observed to coincidewith the direction of the perpendicular to the

maximum principal stress at the critical point in the

llet (i.e. j 25, the origin for the worst crack).3. For all geometries considered, the maximum mode I

stress intensity factor, KI, occurs at the crack angle

a&30, whereas the maximum mode II stressintensity factor, KII, occurs at a&60

.4. For all geometries considered, the mode II stress

intensity factor at crack anglea is equal to the mode I

stress intensity factor at crack angle 90

a.

5. For all geometries, the ratio KI/KII is greater thanunity (mode I dominant) when the crack angle is less

than 458 and less than unity (mode II dominant)

when the crack angle is greater than 458.

6. The ratios KI/KII 0 and ?, which correspond topure mode I and mode II conditions respectively, are

not observed at either a 0 o r 9 08, as might beexpected for simple axial loading. This is due to the

additional shear effect of the applied pressure loading

in the llet region of the components.

7. The difference between KI and KII decreases, for any

crack angle, as the component elastic stress concen-

tration factor decreases.

8. The variation in KI/KII with respect to crack angleis plotted in Fig. 13 for two typical geometries, e.g.


Fig. 12 Variation in KI and KII with crack angle for h=d 0:5, w=d 3 and r=d 0:05 and 0.2 witha 0:5mm and P 100MPa

Fig. 13 Variation in KI=KII with crack angle for h=d 0:5, w=d 1:5 and 3 and r=d 0:05 with a0:5 mm and P

100MPa





12/20

h/d 0.5, r/d 0.05 and w/d 1.5 and 3. The resultsshow that this ratio is w/d independent and is only

crack angle dependent. This ratio can be expressed as

KI=KII paq 14

where the constants p and q can be easily calculated

from Fig. 13, using suitable boundary conditions.

9. KI and KII vary considerably with w/d (greatest

variation), h/dand r/d (least variation) for the range

of geometric parameters considered. This informa-

tion is clearly important from a design viewpoint. KIand KII increase with increasing w/d. This is because

an increase in the projection width and hence the

radial width of the at section of the shoulder (over

which the loading is applied) produces an increase in

the bending component of load, as well as an increase

in the nominal stress. It is also seen that KI and KIIdecrease with increasing h/d. This is because an

increase in the projection length produces an increase

in the bending resistance of the projection section (or

an increase in section modulus of the projection) and

hence provides extra constraints to bending. It is also

observed thatKIand KII increase with decreasing r/d,

i.e. because of an increase in the elastic stress

concentration factor.

Using the KI and KII values obtained from the nite

element predictions, J-integral values have been

obtained using equation (7). The variation in the J

integral with respect to crack angle for some typical

geometries (e.g. h/d 0.5, r/d 0.05 and w/d 1.5 and3) are presented in Fig. 14 for two crack lengths of 0.5

and 2.5 mm. It is observed that the maximum J-integralvalue occurs at a crack angle of a 45 for all

component geometries and all crack lengths considered

in this study.

Mode I and mode II stress intensity factor predictions

for an extensive range of geometric parameters (e.g.

h=d, r=d, w=d), crack angles and crack lengths have beenused to obtain useful prediction equations, using

statistical multiple non-linear regression [13], in the

following form:

KIa Fc,Is0pap 15

KIIa Fc,IIs0pa

p 16

Fc,Iand Fc,II are crack conguration factors correspond-

ing to mode I and mode II situations respectively and

are dened as

Fc,I0:041 0:021 sin p

90a 0:669

0:71 h

d 0:015

0:7

w

d0:008

0:02

r

d0:23

0:4

a

d1:24

17Fc,II a Fc,I 90a 18

The accuracy of equation (17) has been measured by

means of a multiple coefcient of determination, R2,

where 04R2 4 1. This coefcient was found to be

*0.98, which suggests a high level of accuracy of t

between the prediction equation and the nite element

data.

By substituting equations (15) and (16) into equation

(8), an effective crack conguration factor, correspond-

ing to mixed-mode situations, is dened as below. Thiseffective crack conguration factor is referred to as Fc,e,


Fig. 14 Variation in the Jintegral with crack angle for h=d0:5, w=d 1:5 and 3 and r=d 0:05 witha

0:5 and 2.5mm andP

100MPa





13/20

where

JE

p Fc,es0

pa

p 19

and

Fc,e

Fc, I 2 Fc,II 2q

20

It is worth noting that the maximum J value can be

calculated by substituting a

45

in equations (15) to

(20), i.e. Jmax J45. The Jvalue provides an importantcontrolling parameter in the fatigue crack propagation

process under small-scale y ielding conditions. Estimated

values of Fc,e, calculated using equations (15) to (20),

are plotted in Fig. 15 for a typical geometry h=d0:5, w=d 3, r=d 0:05 and crack initiation lengths of0.5 and 2.5 mm. The nite element predictions ofFc,eare

also plotted in Fig. 15. It can be seen that, for both crack

lengths, there is good agreement between the two sets of

data (i.e. within 2 per cent).

5 PREDICTION EQUATIONS BASED ON Kt

5.1 Stress intensity factors

The nite element predictions of the elastic stress

concentration factor, Kt, and the mode I and mode II

stress intensity factors, KI and KII respectively, have

been used to develop useful analytical equations, based

on the statistical multiple non-linear regression model

[13], which can estimate KI and KII, knowing the

component and crack geometries, i.e.

KI 3:25

0:07KI

s0pa

p 0:54

2:23 0:55 sin p90a 0:6

h i

0:15 hd

0:013w

d

0:1 0:017

r

d

0:54a

d

0:1121

and

KII a KI 90a 22

The accuracy of equation (21) has been assessed using a

multiple coefcient of determination [13], R2, where

04R2 4 1. This coefcient was found to be 0.985 in

this case, which demonstrates the accuracy of the model

t to the nite element data.

Equations (21) and (22) can be used for any similar

shaped component and crack geometries within the range

35

h

d5 0:5

35w

d5 1:5

0:25 r

d5 0:05

0:55 a5 2:5 mm

05a5 90

The variation in KI, calculated using equation (21) and

directly from the nite element predictions, with respect

to crack angle is shown in Fig. 16 for a typical geometry

h=d

0:5, w=d

3, r=d

0:05

and crack initiation


Fig. 15 Variation in effective crack conguration factor with crack angle forh=d 0:5, w=d 3 and r=d0:05 with a

0:5 and 2.5mm and P

100MPa





14/20

lengths, a, of 0.5 and 2.5 mm. It can be seen that, in both

cases, there is very good agreement between the two

approaches (within a maximum of 2 per cent).

5.2 J-integral values

In a similar way, J-integral values obtained from thenite element analyses, for an extensive range of

component and crack geometries, have been used to

obtain useful equivalent prediction equations based on

elastic stress concentration factor, using the statistical

multiple non-linear regression method for the same

range of parameters:

Kt5:79

J

s0a

0:417:79 1:79sin

p

90a+ 0:013

h

d

0:017

wd

0:79

rd

0:15

ad

0:057

23

where the negative sign in the last numerator bracket is

used for 455a5 0 and the positive sign is used for905a5 45. In this case, the multiple coefcient ofdetermination, R2, is 0.95, which is more than adequate

at the preliminary design stage. It has been seen that the

maximum J-integral values, Jmax, which are an impor-

tant controlling parameter in fatigue crack propagation

analysis, occurred at a crack angle of a 45. There-fore, an alternative, more accurate, predictive equation

for Jmax, i.e. R2 0.986, is presented:

Kt36:9

Jmax

s0a

0:41h

d

0:013w

d

0:75r

d

0:17a

d

0:068 24

The variation in Jmax, obtained from equation (24) and

directly from the nite element results, with respect tocrack length is plotted in Fig. 17 for typical geometries

h/d 0.5,w/d 1.5 and 3 andr/d 0.05. It can again beseen that in both cases there is very good agreement

between the two methods (within 2 per cent).

For engineering design purposes, an estimation of the

stress intensity factors corresponding to mode I and/or

mode II dominant situations is sometimes demanded.

For example, it has been suggested [18] that, for some

components, mode II has a larger contribution to

fatigue crack growth than mode I or mixed-mode J-

integral values. It has been shown earlier that crack

angles of 30 and 608 are considered to be representativeof the highly dominant mode I and mode II loading

conditions, respectively, for all geometries considered.

Therefore, prediction equations are also presented for

these highly dominant modes. For a 30 or 60,

Kt37:99

J

s0a

0:41h

d

0:013w

d

0:79r

d

0:16a

d

0:066 25

A similar equation has been derived for J-integral


Fig. 16 Variation in KI with crack angle for h=d 0:5, w=d 3 and r=d0:05 with a 0:5 and 2.5mmand P 100MPa





15/20

predictions when a 0 or 908:

Kt44:88

J

s0a

0:41h

d

0:011w

d

0:8r

d

0:132a

d

0:037 26

The accuracy of equations (25) and (26) is demonstrated

by a multiple coefcient of determination, R2 0.99.The J-integral values predicted from these equations are

more accurate than those obtained by substituting

a 0,30,60 and 908 into equation (23). Equations (23)to (26) can be approximated in the following general

form by replacing the exponent 0.41 with 0.5:

J

p FKt

s0a

p 27

where F is a `geometrycrack conguration factor and

is a function of h/d, w/d, r/d and a. By substituting

equation (12) into equation (27), an alternative general

equation for calculating the J-integral values can be

written as

J

p

0:91 h

d

2:11 0:26

w

d

1:44 0:57

r

d

0:92 0:14

h

d

1:68w

d

0:25r

d

0:496F

s0a

p 28

where F can be dened, using equation (27) together

with equations (23), (24), (25) or (26). For example,

when using equation (25) for a crack angle of 30 or 608,

Fbecomes

Fh

d

0:013w

d

0:79r

d

0:16a

d

0:06637:99

29

6 APPLICATION OF RESULTS TO CRACK

INITIATION AND PROPAGATION

Traditionally, fatigue analysis is separated into two

parts, initiation and crack propagation. The initiation

portion of fatigue life consists of crack nucleation

caused by repeated plastic shear straining and a period

of crystallographically oriented crack growth. Propaga-

tion consists of slow stable crack growth followed by

rapid unstable crack growth to nal fracture. Initiation

may be analysed using strain cycle fatigue concepts, and

propagation by linear elastic fracture mechanics con-

cepts. If the majority of the fatigue life is spent in crack

formation and early growth (crack initiation), precise

knowledge of the propagation life is unnecessary for

reasonable estimates of the total fatigue life. Conversely,

when the majority of the fatigue life is spent in crack

propagation, the denition of the initial crack size is

more important than the calculation of initiation life.

However, good estimates of the total life of notch

components, subjected to variable amplitude load

histories, can be obtained if both crack initiation, Ni,

and crack propagation, Np, are considered, i.e.

Nt

Ni

Np

30


Fig. 17 Variation in Jmax with crack length for h=d 0:5, w=d 1:5 and 3 and r=d 0:05 with P 100MPa





16/20

6.1 Crack initiation

As the quest for cost effective nite life designs

continues, there is a n increasing requirement to quantify

the failure performance of components. However, the

conventional methods of achieving this objective (e.g.

prototype testing) are very expensive and time consum-

ing. A number of investigators (e.g. reference [19]) have

suggested alternative approaches based on local strain

and obtained fatigue data from simple uniaxialunnotched specimen tests, where it is assumed that

smooth and notched specimens with the same local

strain range, De, experience the same number of cycles to

fatigue crack initiation,Ni. Smooth specimen fatigue life

data, proposed by MansonCofn, may be expressed in

the following form:

De

2 s

0f

E 2Ni b e0f 2Ni c 31

However, the problem of fatigue crack initiation life

prediction based on a local strain approach becomes oneof estimating the local strain amplitude at the notch.

Local strain amplitude can be determined by prototype

component testing, or can be predicted using nite

element analysis or other numerical or analytical

prediction methods. Prototype testing is very expensive

and time consuming and, although nite element

analysis is very powerful, there are some difculties

when using the method for component design assess-

ments. Therefore, various authors have proposed

analytical relationships for predicting the local strain

amplitude at the root of a notch (see reference [20]).

These relationships, known as notch stressstrainconversion (NSSC) rules, are used to determine the

non-linear and history-dependent stressstrain beha-

viour at the notch root in terms of the load history and

the cyclic deformation properties of the metal. The

commonly used conservative NSSC rules include

Neuber (for plane stress):

DsDe F2nDs0De0 32

where Ds0 and De0 are the nominal stress and strain

range respectively, Ds and De are the local maximum

stress and strain range at the notch, and Fn is a notch

conguration factor, which can be replaced by Kt from

equation (12), for a stepped plate with loaded projectionin order to predict the fatigue crack initiation life of such

a component using the MansonCofn equation [equa-

tion (31)]. The local strain approach associated with the

NSSC rules is a useful and powerful method for

estimating the fatigue crack initiation life of a notched

component. The local strain range is found from the

intersection of equation (32) with the material cyclic

stressstrain curve obtained from smooth specimen

testing:

De

2 Ds

2E Ds

2K0

1=n0

33

By replacing the relevant local strain range in equation

(31), the fatigue crack initiation life,Ni, can be obtained,

as shown in Fig. 18. It is worth recalling that crack

initiation has previously been dened as the formation

of a crack of 0.52.5 mm length.

6.2 Crack propagation

The most widely accepted correlation between constant-

amplitude fatigue crack growth and the applied load is

that suggested by Paris [21] . The rate of crack

propagation per cycle, da=dN, is directly related to themode I cyclic stress intensity, DKI, for uniaxial specimen


Fig. 18 Fatigue crack initiation life prediction procedure based on local strain approach





17/20

testing, in the following way

da

dN C DKI n 34

Fatigue crack growth under mixed-mode loading has

been studied since the 1960s (e.g. reference [22]). The

research has mainly concentrated on two aspects: crack

growth direction and crack growth rate. Many criteria

have been proposed to predict crack growth direction,

including maximum tangential stress, strain energy

density, maximum tangential strain, the T criterion

and the J criterion (e.g. references [10] and [23]). The

parameters proposed for correlating the fatigue crack

growth rate under mixed-mode loading include effective

stress intensity factor, equivalent strain intensity factor

and the J integral (e.g. references [24] a n d [25]). A

detailed review of these theories, including advantages

and limitations, has been presented in reference [26].

6.2.1 Crack growth direction

The criteria considered in this section are based on theJ-

integral approach for the determination of crack growth

direction and crack growth rate. Consider the vector Q

for a two-dimensional elastic crack problem [10] as

Q Q1i Q2j J1i J2j 35where Q1 and Q2 are given as

QjS

Wnj Tkuk,j

dS, j 1 ,2 ,3 36

Wis the strain energy density, nis a normal vector, Tkis

the traction vector,u is the displacement eld, S is every

closed surface bounding a region Rwhich is assumed to

be free of singularities, J1 is equal to the J integral

according to equation (27) and J2 is expressed by

J2G

Wn2 Tkuk,2 ds 37

For a crack in a mixedmode stress eld governed by the

values of stress intensity factors KI and KII, the integral

J1 is given by

J1 c0 18G

K2I K2II 38

The integral J2 can be calculated in a similar way:

J2 c0 1

8G KIKII 39

Consider the projection Q y of the vector Q along adirection making an angley with thex axis, as shown in

Fig. 19, where

J y

J1cos y

J2sin y

40

The initiation of crack growth is governed by the

hypothesis that the crack extends along the radialdirection y yc on which J y becomes a maximum(see Fig. 19). This hypothesis may be expressed

mathematically as

qJ y qy

0, q2J y qy2


18/20

of a suitable model for determining effective stress

intensities that accounts for load ratio, sequence and

crack closure effects needs further work which is

currently in progress for a later publication. However,

a relationship of the form

da

dN ADJB 43

which is analogous to the Paris law [e.g. equation (34)],

has been suggested for crack propagation life predic-

tions under mixed-mode loading conditions [27]. The

value of DJ below which no (measurable) amount of

fatigue crack growth occurs is termed the threshold J

integral, Jth. The implication for design is important

since, if in a cracked structure DJ


19/20

when

DJ JIc 44where Jth and JIc are material constants for a given

thickness under specic environmental conditions.

The critical values of Jth and JIc are related to the

threshold stress intensity factor, Kth, and the critical

strain intensity factor (fracture toughness), KIc, as

follows [10]:

Jth1 v2

E K2th 45

JIc1 v2

E K2Ic 46

7 CRACK PROPAGATION METHODOLOGY

As a typical example, in its simplest form, a fatiguecrack propagation methodology for this plane loaded T-

section under constant-amplitude axial loading can be

summarized as:

1. It is assumed that an initial crack size ofa 0:5 mmis established owing to strain cycle fatigue (described

in section 6.1). The crack angle is assumed to be a 45 (since it corresponds to the maximum J-integralvalue). The remotely applied constant-amplitude

axial nominal stress range, Ds0, is reacted by a

uniform pressure across the entire at section of the

shoulder.

2. The critical crack length, ac, is calculated using

equation (19)JcE

p Fc,eDs0

pac

p 47where JIc is determined from equation (46) and Fc,ecomes from equation 20 [Fc,I and Fc,II can be

calculated from equations (17) and (18) for the given

geometry and with a 45].3. The value ofDJis then calculated, based on equation

(19), i.e.

DJEp Fc,eDs0 pa0p 48

No crack growth occurs if

DJ< Jth 49where Jth is derived from equation (45).

4. If DJ>Jth, the fatigue crack propagation life, Np,can be calculated by integrating the equation

Npac

a0

da

A DJ B 50

where A and B are material constants and DJ is

substituted fromDJE

p Fc,eDs0

pa

p 51Equation (50) should be numerically integrated, as

the value ofFc,e is a function of crack length, a, i.e.

Np 1

A

E

p Ds0

2

!B aca0

da

F2c, ea

B

52

5. The total fatigue life for the component is then given

by equation (30), with the derivation of fatigue crack

initiation life, Ni, as discussed in section 6.1, using

equation (31).

This process can be repeated for any crack angle by

substituting the appropriate value of a into equation

(17).

8 CONCLUSIONS

1. Stress intensity factors for components with loaded

projections can be signicantly higher than the

corresponding values for remote loading, and little

relevant fatigue information is available for this

type of severe loading.

2. In situations where the crack is perpendicular to the

applied loading, a non-zero mode II stress intensity

factor is predicted for these components owing to

the shear effects caused by the pressure loading.

3. The maximum KI value, KII value and J-integral

value occur when the crack originates from a point

around the llet that is at angles ofj&25, 50 and308 respectively to the horizontal.

4. The maximum stress was in the meridional direction

and generally close to the intersection between the

plate and the llet radius, where the Jintegral had a

maximum value (i.e. j&25). On the basis of thisand conclusion 3, results are generally presented for

j 25.5. The J-integral value predictions for this type of

geometry and loading demonstrate good path

independence, and there is good agreement between

values obtained from stresses and those based on

the virtual crack extension method.6. Equation (12) can be used to estimate the notch

conguration (elastic stress concentration) factor

for geometrically similar components with loaded

projections with a high degree of condence.

7. For all geometries considered:

(a) KI is a maximum when the crack angle is

approximately 308;

(b) KII is a maximum when the crack angle is

approximately 608;

(c) KI KII when the crack angle is 458;(d) KI for a is the same as KII for 90

a;






20/20

(e) KI and KII increase with decreasing h/d, increas-

ing w/dand decreasing r/d;

(f) the maximum J-integral value occurs when the

crack angle is 458.

8. Equations (17) and (18) can be used to estimate the

mode I and mode II crack conguration factors for

geometrically similar components with loaded

projections with a high degree of condence.

9. Equations are presented that can provide estimatesfor crack initiation and propagation lives without

the need to carry out extensive nite element linear

elastic fracture mechanics an alyses.

10. Equations (21) and (22) can be used to estimate

mode I and mode II stress intensity factors from the

elastic stress concentration factor and the crack

angle for at bars a nd plates with load ed projections

with a high level of condence.

11. Similarly, equation (23) can be used to estimate the

J-integral value for any crack angle with reasonable

accuracy. Greater accuracy can been achieved by

using equations (24) to (26) for the specic crackangles of 0 and 908 (which represent the extreme

mode I and mode II cases), 30 and 608 (which are

the dominant mode I and mode II angles respec-

tively) and 458 (at which the maximum J-integral

value is predicted).

12. The direction in which a crack will grow was not

found to vary signicantly with initial crack angle,

crack length or component geometry.

13. The methodology presented in section 7 can be

easily used to estimate the fatigue life of such

components with a knowledge of the component

geometry a nd load ing, together with the initial crackangle and length.

REFERENCES

1 Hertzberg, R. W. Deformation and Fracture Mechanics of

Engineering Materials, 1976 (John Wiley, New York).

2 Paris, P. C. FatigueAn Interdisciplinary Approach, 1964

(Syracuse University Press, Syracuse, New York).

3 Paris, P. C.and Sih, G. Fracture toughness testing and its

application. In ASTM STP 381, 1965, pp. 3081 (American

Society for Testing and Materials, Philadelphia, Pennsyl-

vania).

4 Harris, D. O. Stress intensity factor for hollow circumfer-

entially notch round bars.J. Basic Engng, 1967,89, 4954.

5 Hellen, T. K. On the method of virtual crack extensions.

Int. J. Numer. Meth. Engng, 1975, 9, 187207.

6 Engineering Sciences Data Unit, data sheet 69020.

7 BS 5447 Methods of Test for Plane Strain Fracture

Toughness (KIc) of Metallic Material, 1977 (British

Standards Institution, London).

8 ANSYS, Version 6.1, Data Preparation Manual, 1973

(ANSYS, Canonsburg, Pennsylvania).

9 Gray, T. G. F., Tournery, F. and Spence, J. Analysis of

t t ti f t f t d l t b d

crack tip stress intensity approach. J. Strain Analysis, 1996,

31(3), 197204.

10 Gdoutos,E. E. FractureMechanicsCriteriaand Applications,

1990 (Kluwer A cademic Publishers, Boston and London).

11 Tada, M., Paris, P. and Irwin, G. The Stress Analysis of

Cracks Handbook, 1973 (Del Research Corporation,

Hellertown, Pennsylvania).

12 Peterson, R. E. Stress Concentration Design Factors, 1974

(John Wiley, New York).

13 Gowhari-Anaraki, A. R.andHardy, S. J. Design equations

for elastic stress concentration factors in hollow tubes with

axisymmetric internal projections subjected to axial load-

ing. J. Strain Analysis, 2001, 36(5), 531534.

14 Singh, S.Machine Design, 1999 (Khanna Publishers, Delhi,

India).

15 Socie, D. F. Fatigue life estimates for bluntly notched

members.J. Engng. Mater. Technol., 1980,102, 153158.

16 Dowling, N. E. Crack growth during low-cycle fatigue of

smooth axial specimens. In Cyclic StressStrain and Plastic

Deformation Aspects of Fatigue Crack Growth, ASTM STP

637, 1977, pp. 97121 (American Society for Testing and

Materials, Philadelphia, Pennsylvania).

17 Haddad, M. H., Smith, K. N. and Topper, T. H. Fatigue

crack propagation of short cracks.Trans. ASME, J. Engng

Mater. Technol., 1979,10 1(1), 573580.

18 Smith, E. W.and Pascoe, K. J. Fatigue crack initiation and

growth in a high strength ductile steel subject to in-phase

biaxial loading. In ASTM STP 853, 1985, pp. 111134

(American Society for Testing and Materials, Philadelphia,

Pennsylvania).

19 Topper, T. H. an d Gowda, C. V. B. G. Local stressstrain

approach to fatigue analysis and design. In ASME Design

Engineering Conference, Chicago, Illinois, 1970, paper 70-

DE-24.

20 Hardy, S. J. and Gowhari-Anaraki, A. R.Stress and strain

range predictions for axisymmetric and two-dimensional

components with stress concentrations and comparisons

with notch stressstrain conversion rule estimates.J. Strain

Analysis, 1993,28(3), 209221.

21 Paris, P. C. The fracture mechanics approach to fatigue. In

Proceedings of 10th Sagamore Conference, 1963 (Syracuse

University Press, Syracuse, New York).

22 Iida, S. and Kabayashi, A. S. Crack propagation rate in

7075-T6 plates under cyclic tensile and transverse shear

loading.Trans. ASME, J. Basic Engng, 1969,91, 764769.

23 Theocaris, P. S., Kardomateas, G. A. and Andrianopoulos,

N. P. Experimental study of the T-criterion in ductile

fracture.Engng Fracture Mechanics, 1982,17, 439447.

24 Tanaka, K. Fatigue crack propagation from a crack

inclined to the cyclic tensile axis. Engng Fracture

Mechanics, 1974, 6, 493507.

25 Socie, D. F., Hua, C. T.and Worthem, D. W. Mixed mode

small crack growth. Fatigue and Fracture of Engng Mater.

Struct., 1987,10(1), 116.

26 Bold, P. E., Brown, M. W. and Allen, R. J. A review of

fatigue crack growth in steels under mixed mode I and II

loading. Fatigue and Fracture of Engng Mater. Struct.,

1992,15(10), 965977.

27 Begley, J. A. a ndLandes, J. D. The J-integral as a fracture

criterion. In Fracture Toughness, ASTM STP 514, 1972,

pp. 123 and 2439 (American Society for Testing and

M t i l Phil d l hi P l i )



Mixed-Mode Fatigue Crack Propagation in Thin

Documents

Transcript of Mixed-Mode Fatigue Crack Propagation in Thin