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6_x_wang_2009

Determination of approximate point load weight functions for embedded elliptical cracks

Xin Wang a,*,Grzegorz Glinka b

a Department of Mechanical and Aerospace Engineering,Carleton University,Ottawa,Ontario,Canada K1S 5B6b

Department of Mechanical Engineering,University of Waterloo,Waterloo,Ontario,Canada N2L 3G1

a r t i c l e i n f o Article history:

Received 29August 2008

Received in revised form 4December 2008Accepted 9December 2008

Available online 24December 2008Keywords:

Embedded elliptical crack Weight function

Stress intensity factor

Two-dimensional stress ?elds

a b s t r a c t

This paper presents the application of weight function method for the calculation of stress intensity fac-tors in embedded elliptical cracks under complex two-dimensional loading conditions.A new general mathematical form of point load weight function is proposed based on the properties of weight functions and the available weight functions for two-dimensional cracks.The existence of this general weight func-tion form has simpli?ed the determination of point load weight functions signi?cantly.For an embedded elliptical crack of any aspect ratio,the unknown parameters in the general form can be determined from one reference stress intensity factor solution.This method was used to derive the weight functions for embedded elliptical cracks in an in?nite body and in a semi-in?nite body.The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions.The derived weight functions are particularly useful for the fatigue crack growth analysis of planer embedded cracks subjected to ?uctuating non-linear stress ?elds resulting from surface treatment (shot peening),stress concentration or welding (residual stress).

1.Introduction

A problem frequently encountered in applied fracture and fati-gue analysis is the calculation of the stress intensity factors for defective components subjected to a complex stress distribution.Weight function method [1,2]has been widely used in the deter-mination of stress intensity factors for its distinctive advantage of separating the loading and the geometry.Once the weight func-tion is known for a given cracked geometry,the stress intensity factor due to any load system applied to the body can be deter-mined by using the same weight function.Therefore it is especially suited when a large number of stress intensity factor solutions are desired for complex stress distributions.The success of the weight function method depends on the accurate determination of the weight function itself.Because the concept of ‘‘weight function”was originally introduced [1,2]for one-dimensional edge cracks or through cracks,most of work in the literature has been concen-trated on one-dimensional cracks.The methods of obtaining the weight functions for one-dimensional cracks have been well devel-oped,see [3–5]for example.However,to date,the method of obtaining weight functions for two-dimensional cracks is still not well developed.For two-dimensional cracks,close-form exact weight functions are available only for very limited cases:the circular crack and the half plane crack in an in?nite body [6].

Embedded elliptical crack is one of the most used models for two-dimensional embedded cracks in many engineering compo-nents (Fig.1).Fig.1a shows an embedded elliptical crack in an in?-nite body (i.e.,b ,t and h )a and c );and Fig.1b illustrates an embedded elliptical crack in a semi-in?nite body,where the embedded crack in close to one free surface represented by the dis-tance d .The notation of the ellipse geometry is shown in Fig.2.For embedded elliptical crack in an in?nite body (Fig.1a),stress inten-sity factor solutions are available for limited simple loading condi-tions.When the uncracked stress distribution in the area to be occupied by the elliptical crack is a simple one such as uniform (tension)or one-dimensional linearly varying (bending),the well-known explicit solutions [7,8]can be used to determine the stress intensity factor at any point,P 0,along the crack front.When the stress distribution is two-dimensional depending on two vari-ables,r (x ,y ),which is the case in many engineering applications,more complicated calculations have to be made.Exact stress inten-sity factor solutions for 2D polynomial stress distributions of the order of three were provided by Shah and Kabayashi [9],and for polynomial of any order n were provided by Vijaykumar and Atluri [10]and Nishioka and Atluri [11],where tedious labour intensive evaluation of elliptic integrals are involved.

For embedded elliptical cracks in a semi-in?nite body (Fig.1b),the effect of free boundary needs to be included and there is no analytical solution https://www.wendangku.net/doc/1718573447.html,ing ?nite element method,the stress intensity factors were obtained by Shiratori et al [12]for polynomial stress distributions of order 3.For complex stress

*Corresponding author.Tel.:+16135202600x8308;fax:+16135205715.E-mail address:xwang@mae.carleton.ca (X.Wang).International Journal of Fatigue 31(2009)

1816–1827

Contents lists available at ScienceDirect

International Journal of Fatigue

journal homepage:w w w.e l s e v i e r.c o m/l o c a t e /i j f a t i g u

distributions which can be approximated using polynomials of or-der3,the stress intensity factor can be then calculated using the superposition method.However,there are stress distributions which cannot be easily represented by polynomials.It is therefore of practical signi?cance to develop weight function solutions for both embedded elliptical cracks in in?nite body(Fig.1a)and in semi-in?nite body(Fig.1b),which will enable the determination of stress intensity factor for any complex stress distributions.

Several attempts have been made to derive the weight func-tions for embedded elliptical crack by solving the problem analyt-ically,see recent development in[13,14].However,these solutions are in a series expansion form and cannot be easily used for engi-neering applications.Numerical methods were developed that can be used to determine the weight functions numerically,for exam-ple using?nite element method[15,16].However,the amount of work needed to determine the weight function for two-dimen-sional cracks with numerical method can be very large,and the weight functions in numerical form are often inconvenient to use.

One of the important features of weight function method is that the weight functions for a wide variety of crack con?gurations can be represented using the same mathematical form,which has been demonstrated extensively from weight function theories for one-dimensional crack problems[5].The existence of a general mathe-matical form simpli?ed the determination of weight functions sig-ni?cantly.Several attempts have been made to?nd the general weight function form for two-dimensional crack.The most com-monly used method is the O-integral method[17,18].However, it has been realised that the O-integral functional form works well for some geometries but its accuracy is not as good for other geom-etries,particular for low aspect ratio embedded elliptical cracks [18].Various analytical methods have also been used by several authors to develop approximate weight functions speci?cally for embedded elliptical cracks[19–21].However,no general accepted form has been established for2D cracks.

The objective of the present paper is the development of approximate weight functions for embedded elliptical cracks based on a newly proposed general form for two-dimensional crack prob-lems.The paper is organized as follows.First,the background of weight function method is reviewed,and a general form for point load weight function is proposed and the method of?nding the parameters of the weight function is discussed.Based on the devel-oped method,weight functions for embedded elliptical cracks in an in?nite body(Fig.1a)and in a semi-in?nite body(Fig.1b)for a wide range of aspect ratios of a/c(see Fig.2)is derived and vali-dated against available stress intensity factor solutions for several linear and non-linear stress distributions.

2.Approximate point load weight functions

2.1.Theoretical background

The weight function technique for calculating stress intensity factors is based on the principle of superposition.For one-dimen-sional cracks,it can be shown[1]that the stress intensity factor for a cracked body(Fig.3a)subjected to the external loading sys-tem,S,is the same as the stress intensity factor in a geometrically identical body(Fig.3c)with the local stress?eld r(x)applied to the crack faces.The local stress?eld,r(x),induced in the prospective crack plane by preload S,is determined from an uncracked body (Fig.3b).The stress intensity factor for a cracked body with loading applied to the crack faces can be calculated by integrating the product of the weight function,m(x,a),and the stress distribution, r(x),in the crack plane:

Z a

rexTmex;aTd xe1T

The weight function m(x,a)depends only on the geometry of the crack and the cracked body.Once the weight function has been determined,the stress intensity factor for this geometry can be ob-tained from Eq.(1)for any stress distribution,r(x).Mathemati-cally,the weight function,m(x,a),is the Green’s function for the present boundary value problem scaled with respect to the crack dimension a.It represents the stress intensity factor at the crack tip for a pair of unit point loads acting on the surface at the location x.

For a two-dimensional crack,the stress intensity factors vary along the crack front,as shown in Fig.4.The counterpart to Eq.

(1)for two-dimensional cracks is a double integral over the crack surface

KeP0T?

Z Z

rex;yTmex;y;P0Td Se2T

Nomenclature

A,B,C three points along the crack front for embedded crack in semi-in?nite body

a half length of minor axis for embedded elliptical crack;

crack length

b,t,h dimensions of the cracked body

c half length of major axis for embedde

d elliptical crack

d distanc

e o

f embedded crack in semi-in?nite body to free

surface

E Young’s modulus;shape factor for an ellipse

F normalized stress intensity factor

H generalized elastic modulus

K mode I stress intensity factor

K r reference stress intensity factor

l small crack front segment variation

m(x,y;P0)point load weight function

M i,M parameters in point load weight function expression

P point load location on crack face

P0point along the crack front under consideration r radius of polar coordinate

S entire crack face area

R radius of polar coordinate for points along crack front s shortest distance between point load and the boundary of crack front

u r crack face displacement for reference stress intensity factor

x,y Cartesian coordinates for point P

w(x,y;P0)regular function in weight function expression

h angle of polar coordinate for crack front point P0

q distance between load point P and point along crack front under consideration P0

/parametric angle of ellipse for point P0

m Poisson’s ratio

r0nominal stress

a aspect ratio of elliptical crack=a/c

C boundary of crack front

u angle of polar coordinate for load point P

X.Wang,G.Glinka/International Journal of Fatigue31(2009)1816–18271817

where m (x ,y ;P 0)is the point load weight function,it represents

stress intensity factor at point P 0on the crack front for a pair of unit point loads acting on the crack surface at point (x ,y )as shown in Fig.4,and r (x ,y )is a general two-dimensional stress distribution as shown in Fig.4.If the stress distribution r (x ,y )is one-dimen-sional,for example,only a function of x ,then Eq.(2)can be simpli-?ed to

K eP 0

r ex T

m ex ;y ;P 0

Td y d x ?

r ex TM ex ;P 0Td x

e3T

where M (x ;P 0)represents the stress intensity factor at point P 0for

unit line load at position x as shown in Fig.4;and a is the crack depths in the x -direction.In other words,M (x ;P 0)is the line load weight function for two-dimensional cracks.For any one-dimensional or two-dimensional cracks,if the weight functions m (x ,a ),m (x ,y ;P 0)or M (x ;P 0)are obtained,the stress intensity factors for other loading conditions can be calcu-lated using Eqs.(1)–(3).

For one-dimensional cracks,the determination of the weight function,m (x ,a ),in Eq.(1)can be simpli?ed by using the relation between stress intensity factor under consideration and a refer-ence stress intensity factor solution and the corresponding crack face displacement,as derived in [2]

K ?K r ?H

r ex T

o u r ex ;a T

o a

d x e4T

where H is the generalized elastic modulus which equals E for plane stress or E /(1àm 2)for plane strain,and K and u r are the stress inten-sity factor and corresponding crack face displacement for one refer-ence stress distribution.From Eq.(4),the weight function m (x ,a )can be obtained as

m ex ;a T?

H K r o u r ex ;a To a

e5T

Eq.(5)provides an ef?cient way to determine weight function from a reference stress intensity factor solution and the corresponding displacement ?elds.An appropriate reference stress intensity factor K r can often be found either in the literature or by numerical calcu-lation.Although the corresponding analytical expression for the crack opening displacement function u r (x ,a )is more dif?cult to ob-tain,because it is seldom published together with stress intensity factor solutions,several authors (Petroski and Achenbach,[22];Wu and Carlsson [3];Fett and Munz [4];Glinka and Shen [5])have proposed approximate expressions for the displacement,u r (x ,a ),or the weight function,m (x ,a ).Glinka and Shen [5]have found that the mode I weight functions for a variety of 1D crack geometries can be accurately approximated using the following expression

m ex ;a T?2?????????????????????2p ea àx T

p 1tM 11à

x a 1=2

tM 21àx a tM 31àx a

3=2

e6T

As shown in Eq.(6),the weight function has the same singular term and M 1–M 3are parameters of the non-singular term and they depend on the speci?c crack geometry.The existence of a general weight function form simpli?ed the determination of weight

B Fig.1.Elliptical crack in an in?nite solid (a)and in a semi-in?nite body

(b).

1818X.Wang,G.Glinka /International Journal of Fatigue 31(2009)1816–1827

functions;the derivation of weight function for a particular geo-metrical con?guration of cracked body is reduced to the determi-nation of parameters M 1–M 3.

For two-dimensional cracks (see Fig.5),the counter-part rela-tionship between two stress intensity factor solutions (Eq.(4))are also derived by Rice [2,6]

C ?K ?K r d l d C ?H

Z Z

r ex ;y Td u r ex ;y Td S

e7T

where C represents the crack front,d l is a smooth function of posi-tion along C making the advance of the crack in a direction locally normal to C .d u (x ,y )is the ?rst order variation of displacement cor-responding to d l .If we de?ne the d l is local to point P 0,and the cor-responding area change is d F P 0=d l ?d C ,the weight function m (x ,y ,P 0)can be obtained

m ex ;y ;P 0T?

H K r eP 0Td u r ex ;y Td F P 0

e8T

That is if the three-dimensional solutions of any reference load system is known so that the ?rst order variation d u (x ,y )can be determined,at any point along the crack front P 0,corresponding to variation d F P ,the weight function for point P 0can be obtained from Eq.(8).

Several methods were developed to apply Eqs.(7)and (8)to de-rive weight functions for two-dimensional cracks.However,the complete solution for d u (x ,y )for arbitrary d F P is much more dif?-

culty to obtain than o u r /o a in the one-dimensional crack problem,several simpli?cations were used (See Fett [23],for example).On the other hand,the line load weight function for two-dimen-sional cracks M (x ;P 0)in Eq.(3)can be determined in a similar way as deriving weight function m (x ,a )for one-dimensional cracks.For surface cracks and corner cracks,the general weight function forms of M (x ,P 0)have been obtained for the deepest point M (x ,A 0),surface point M (x ,B 0)and general point M (x ,P 0)[24,25].Using these general forms,the line load weight functions were derived for a variety of two-dimensional crack geometries [24–28].

In spite of the high ef?ciency and usefulness of the line load weight function in engineering applications,they cannot be used if the stress ?eld is of a two-dimensional nature,i.e.,when the stress ?eld r (x ,y )depends on both x and y coordinates.Therefore,a general method for the determination of point load weight func-tion m (x ,y ;P 0)is needed.

2.2.Properties of point load weight functions for two-dimensional cracks

By analyzing the properties of weight functions for two-dimen-sional cracks,Rice [6]pointed out that,for an embedded planar crack in an in?nite body (see Fig.5),there are two key parameters s and q in the weight function expression,m (x ,y ;P 0).Here s is the

σ(x, y )

Crack Surface S

P’

Fig.4.Two-dimensional crack under two-dimensional stress distribution.

P(x, y)

P’(x’, y’)s

Fig.5.Weight function notation for general two-dimensional crack.

X.Wang,G.Glinka /International Journal of Fatigue 31(2009)1816–18271819

shortest distance between the load point P (x ,y )and the boundary of the crack front C ,and q is the distance between the load P and the point P 0under consideration (see Fig.5).These two param-eters can be used to describe available analytical weight functions.For the half-plane crack in an in?nite body as shown in Fig.6a,the weight function is

m ex ;y ;P 0

?????2s

p p q e9T

For the penny shaped crack as shown in Fig.6b

m ex ;y ;P 0

T??????2s p p 3=2q

2??????????????1à

2a r e10T

where a is the radius of the penny shaped crack.

For an arbitrary planar crack embedded in an in?nite body (Fig.5),the weight function can be represented using the following general expression [6]

m ex ;y ;P 0T?

?????2s

p p 3=2

w ex ;y ;P 0T

e11T

It is apparent from Eq.(11)that the singularity term in the point load weight function is of the order of ??

p q

2,and weight function tends to in?nite when q approaches zero.When s equals zero and q is not zero,the weight function value is zero.

The function w (x ,y;P 0)describes the effect of the embedded crack geometry con?guration.For any given crack geometry,if the function w (x ,y;P 0)can be obtained,then the point load weight function can be obtained from Eq.(11).For the half plane crack

w ex ;y ;P 0

T?1

e12T

and for the penny shape crack

w ex ;y ;P 0

T???????????????

1à

r e13T

It was also pointed out by Rice [6]that the function w (x ,y;P 0

)

has a well-de?ned limit when point P (x ,y )approached the crack boundary,i.e.,s approaches 0.For both cases of a half plane or pen-ny shaped crack

lim s !0

?w ex ;y ;P 0T ?1

e14T

That is the function w (x ,y ;P 0)in Eq.(11)is a regular function (without any singularity),and the singular term has already been

represented by the term ??s

p q

Note that as shown in Eq.(8),the weight function is closely re-lated to the crack opening displacement ?elds.It has also been shown by Bueckner [1]that weight functions are in fact singular

displacements of so-called ‘‘fundamental ?elds,”which produce an in?nite energy in a small volume.

2.3.Proposed general form for point load weight function

Now consider the embedded elliptical crack as shown in Fig.7.The objective is to ?nd the weight function m (x ,y ,P 0).Based on the preceding discussion,it is reasonable to further represent the point load weight function using the following form:

m ex ;y ;P 0T??????2s p p 3=2q 21tX n i ?1

M i eh ;a T1à

r eu T

R eu T i "#e15T

i.e.,

w ex ;y ;P 0T?1t

X n i ?1

M i eh ;a T1à

r eu T

u i "

e16T

That is:the weight function can be represented by the summa-tion of two parts;the ?rst part is the singular term,and the second part accounts for the effect of crack con?gurations.Here h repre-sents the location of P 0;and r ,u are the polar coordinates of point P (x ,y ).And R (u )is the corresponding point on the crack front (See Fig.7).Note that the weight function parameter M i depends on the aspect ratio of the ellipse,a =a /c .

It can be easily shown it satis?es the condition represented by Eq.(14).In addition,for the special case of half plane crack,from Eq.(12),we have

M 1?M 2?M 3ááá?0e17T

P(x, y)

P(x’, y’)

P(x, y)

ρs

P’(x’, y’)

Fig.6.Weight function for (a)half plane crack and (b)circular crack.

P’(x’, y’)

P(x, y)

OP = r(φ)

OQ = R(φ)

Fig.7.Weight function for embedded elliptical crack.

1820X.Wang,G.Glinka /International Journal of Fatigue 31(2009)1816–1827

for circular crack,the exact weight function Eq.(13)gives

w ex ;y ;P 0

T?1às 2a 1=2?1à12e1àr

1=2e18T

It can be represented using a following series expansion (i.e.,Eq.

(15))

w ex ;y ;P 0T?1tM 11àr R tM 21à

r R 2tM 31àr R

tááá

e19T

In fact,further studies indicate,which will be discussed in the following Section 2.4,that only two-terms of Eq.(19)are required to provide good approximation of Eq.(18)with M 1(h ,a )=à0.23213and all other M i =0(i =2,3and ...).

Another available weight function for cracks in an in?nite body is the tunnel crack geometry.Through an approximate analysis,the weight function was presented in [18]for tunnel crack in an in?-nite body,which can be further accurately approximated using Eq.(19).

In summary,our analyses have indicated that the general form of Eq.(15),with one term,i.e.,n =1,can approximate the point load weight functions with good accuracy for a wide range of embedded crack con?gurations.That is

m ex ;y ;P 0

T??????2s p p 3=2q

21tM eh ;a T1à

r eu T

R eu T e20a Tor w ex ;y ;P 0

T?1tM eh ;a T1à

r eu TR eu T

e20b T

In addition,for the case of embedded elliptical crack in a semi-in?nite body (Fig.1b),the weight functions will also depend on the distance of the crack to the free surface,d .This effect can be ac-counted for by the M factor in Eq.(20).It is also a function of the distance.It is therefore reasonable to further represent the corre-sponding weight function as

m ex ;y ;P 0

T??????2s p p 3=2q

21tM h ;a ;d a 1à

r eu T

R eu T e21a Tor w ex ;y ;P 0

T?1tM h ;a ;d a 1àr eu TR eu T

e21b T

The derivation of point load weight function can then be simpli-?ed using this general expression.Here d /a is the non-dimensional

distance.

2.4.Determination of weight function parameters

Knowing the general weight function form,Eq.(21)or (20)which is a speci?c case of (21)),the derivation of weight function for a particular embedded elliptical crack has been reduced to the derivation of parameters M (h ,a ,d /a )along the entire crack front.

The parameter M (h ,a ,d/a )can be determined using Eq.(2)pro-vided that one reference stress intensity factors solution K r is known.The stress distribution expression and the general weight function expression Eq.(21)can be substituted for r (x ,y )and m (x ,y ;P 0)into Eq.(2).This leads to the equation for the determina-tion of the unknown parameter M (h ,a ,d /a )

K r eP 0

Z Z

r r ex ;y T?????2s

p p 3=2q

21tM h ;a ;d a 1àr eu TR eu T

e22T

After integration,Eq.(22)can be used to solve for M (h ,a ,d /a ).Note

that this calculation needs to be carried out at every point along the crack front.The M (h ,a )in Eq.(20)can be determined in the exact same way,except it is independent of d/a

Fig.8.Typical mesh used for numerical integration (a /c =0.2).

Table 1

Weight function parameter M (/,a )for a =0.1,0.2,0.4,0.6,0.8and 1.2//p a =0.1

a =0.2

a =0.4

a =0.6

a =0.8

a =1

à4.35à2.1772à0.9053à0.5042à0.3262à0.23210.0625à3.5898à1.9917à0.8735à0.4958à0.3241à0.23210.125à2.5633à1.6044à0.7895à0.4721à0.3178à0.23210.1875à1.7968à1.2214à0.6784à0.4371à0.3079à0.23210.25à1.2669à0.9121à0.5634à0.3954à0.2952à0.23210.3125à0.9127à0.6811à0.4587à0.3518à0.2806à0.23210.375à0.6774à0.5144à0.3701à0.3096à0.2651à0.23210.4375à0.5192à0.3959à0.2986à0.2711à0.2494à0.23210.5à0.4107à0.3117à0.2423à0.2374à0.2343à0.23210.5625à0.3346à0.2515à0.199à0.2089à0.2202à0.23210.625à0.2801à0.2083à0.166à0.1853à0.2076à0.23210.6875à0.2404à0.177à0.1412à0.1664à0.1966à0.23210.75à0.2114à0.1544à0.123à0.1516à0.1875à0.23210.8125à0.1904à0.1384à0.11à0.1407à0.1803à0.23210.875à0.1757à0.1277à0.1013à0.1331à0.1751à0.23210.9375à0.1668à0.1214à0.0963à0.1287à0.172à0.23211

à0.1636

à0.1192

à0.0946

à0.1271

à0.1709

à0.2321

X.Wang,G.Glinka /International Journal of Fatigue 31(2009)1816–18271821

3.Point weight function for embedded elliptical cracks

In this section,Eq.(22)is applied to derive weight functions for embedded elliptical cracks in both in?nite and semi-in?nite bodies.The M factors are determined for each crack con?gurations. The derived weight functions are then validated using stress inten-sity factor solutions for other loading conditions.For any point P0 along the crack front,it can either be identi?ed by the polar angle h or parametric angle/(see Figs.2and7),and they are related simply by:tan h=a tan/.In the following sections,parametric an-gle/is used to represent point P0.

3.1.Embedded elliptical cracks in an in?nite body

The weight functions for embedded elliptical crack in an in?nite body(Fig.1a)in the format of Eq.(20)are derived and validated in this section.The solutions are derived for the entire crack front.

3.1.1.Reference stress intensity factor solutions

For embedded elliptical crack as shown in Fig.1a,the stress intensity factor for uniform stress?eld is used as reference solu-tion.The uniform stress is applied directly on to the crack face

rex;yT?r0e23TThe exact stress intensity factor solution is[7]

Ke/T?r

??????p

sin2/t

a2cos2/

1=4

e24T

where/is the parametric angle,a=a/c and E is the complete ellip-tical integral of second kind,given by the following empirical equa-tion[29]

???????????????????????????????????????

1:0t1:464eaT1:65

e25T

3.1.2.Determination of weight functions

By substituting Eqs.(23)and(24)into Eq.(2),an equation with unknown M(/,a)is established.Numerical integration is required to solve for M(/,a).

A computer program was developed to perform the numerical integration based on the standard Gauss-Legendre quadrature technique.Curved eight-nodes elements are used to distretize the entire elliptical areas.The integration algorithm was veri?ed using the analytical weight function for a penny shaped crack (a/c=1).For uniform stress?eld,the maximum difference be-tween the exact stress intensity factor solution and the calculation based on the present integration routine was less than0.8%along the whole crack front.Fig.8shows a typical mesh used in the pres-ent calculations for a/c=0.2.

The results for the parameters M(/,a)are obtained and pre-sented in Table1for points along the crack front.The aspect ratios considered are a/c=0.1,0.2,0.4,0.6and0.8.The results are plotted in Fig.9.Note the function M(/,a)is symmetric about both x and y axis,therefore only a quarter(06/6p/2)is https://www.wendangku.net/doc/1718573447.html,ing this method,the M(/,a)for penny shaped crack is found to be à0.23213.For comparison,the results for penny shaped crack (a/c=1)are also presented.For engineering applications,the M(/,a)factors are?tted using empirical formulas,and it is sum-marised in Appendix A1.

3.1.3.Validation of weight functions

Six different loading cases were applied to the surface of the elliptical crack to validate the derived weight functions in the form of Eq.(20).Applying Eq.(2),stress intensity factors along the crack front of an embedded elliptical crack of aspect ratio a=0.1,0.2,0.4, 0.6and0.8were calculated for the following stress?elds

Uniform stress?eld

rex;yT?r0e26TOne-dimensional linear stress?eld depending on coordinate x

rex;yT?r0x

e27TOne-dimensional linear stress?eld depending on coordinate y

rex;yT?r

e28TTwo-dimensional non-linear stress?eld

rex;yT?r0xy

e29T

1822X.Wang,G.Glinka/International Journal of Fatigue31(2009)1816–1827

One-dimensional quadratic stress ?eld depending on coordinate x

r ex ;y T?r 0

x a

2e30T

and one-dimensional quadratic stress ?eld depending on coordinate y

r ex ;y T?r 0

y c

e31T

The resulting stress intensity factors were normalized as follows:

F e/T?K e/Ter 0??????p a p =E T

e32T

where F is the boundary correction factor,and E is given in Eq.(25).

The boundary correction factors F from the weight function pre-dictions were compared with the exact solutions for these six load-ing conditions [9].The results are shown in Figs.10–14for aspect ratios a =0.1,0.2,0.4,0.6and 0.8,respectively.Note that the uni-form stress distribution is the reference case.For other linear and non-linear loadings,the differences were generally within 5%for all the aspect ratios.These accuracies are in the same range as the weight functions developed in [21].But the present weight function is much easier to implement.

3.2.Embedded elliptical cracks in an semi-in?nite body

In this section the weight functions for embedded elliptical crack in a semi-in?nite body (Fig.1b)are derived and validated.

X.Wang,G.Glinka /International Journal of Fatigue 31(2009)1816–18271823

The weight functions are derived using one reference stress intensity factor solution corresponding to uniform loading condition,and validated using several linear and non-linear stress distributions.

3.2.1.Reference stress intensity factor solutions

The stress intensity factors at the three points A,B and C along the crack front(see Figs.1b and2)were obtained using?-nite element method in Ref.[12].The uniform stress was applied directly on to the crack face(Eq.(23)).The stress intensity factor was normalised following Eq.(32).The resulting F for a/c=0.2, 0.4,0.6and1.0and d/a=0.25,0.4,0.5and0.625are summarised in Table2.They are used as the reference stress intensity factor solutions.3.2.2.Determination of weight functions

Following the same procedure as Section3.1.2,the weight func-tion is now in the format of Eq.(21).The factor M(h,a,d/a)is solved at point A(corresponds to/=àp/2),B(/=p/2)and C(/=0).The results for the parameters M(/,a,d/a)are obtained and presented in Table3.Empirical formulas of M(/,a,d/a)are given in Appendix A2.Note the geometry is only symmetric about y axis,and M at point A and B are therefore have different values.

3.2.3.Validation of weight functions

The derived weight functions for point A,B and C are validated using?nite element results for various linear and non-linear stress https://www.wendangku.net/doc/1718573447.html,ing Eq.(2),stress intensity factors are calculated for the following stress?elds applied to the crack face

1824X.Wang,G.Glinka/International Journal of Fatigue31(2009)1816–1827

r ex ;y T?r 01à

x a

n ?0;1;2;3e33T

The stress intensity factors calculated from weight functions are compared to the ?nite element results from [12]for the same stress

Table 2

Reference stress intensity factor solutions at point A (/=àp /2),Point B (/=p /2)and Point C (/=0),from Ref.[12].

a /c

d/a =0.25d/a =0.4d/a =0.5d/a =0.625Point A

0.2 1.355 1.238 1.195 1.1570.4 1.271 1.174 1.138 1.1080.6 1.209 1.131 1.102 1.0861.0 1.143 1.089 1.071 1.055Point B

0.2 1.108 1.088 1.079 1.0710.4 1.078 1.063 1.056 1.0480.6 1.053 1.043 1.038 1.0331.0 1.041 1.038 1.037 1.035Point C

0.20.4670.4640.4620.4600.40.6850.6760.6720.6680.60.8150.8190.8140.8101.0

1.050

1.040

1.036

1.031

Table 3

Weight function parameter M (/,a ,d/a )for a =0.2,0.4,0.6and 1and d/a =0.25,0.4,0.5and 0.625.

a /c

d/a =0.25d/a =0.4d/a =0.5d/a =0.625

Point A

0.20.89410.56010.43730.32890.40.78900.47270.35530.25750.60.61690.33920.23600.17901.00.33860.11870.0454à0.0196Point B

0.20.18900.13190.10620.08340.40.15970.11080.08790.06180.60.06160.02590.0081à0.0091.0à0.0766à0.0888à0.0929à0.1174Point C

0.2à1.8246à1.8780à1.9137à1.94930.4à0.4171à0.4727à0.5379à0.57500.6à0.2463à0.2207à0.2526à0.27821.0

à0.0400

à0.0808

à0.0971

à0.1011

X.Wang,G.Glinka /International Journal of Fatigue 31(2009)1816–1827

1825

distributions.Very good agreements were achieved for all the as-pect ratios,a /c =0.2,0.4,0.6and 1.0and d /a =0.25,0.4,0.5and 0.625at all these points.The maximum differences are within 5%.The boundary correction factor results F for d/a =0.25are shown in Figs.15–17for points A,B and C,respectively.

4.Conclusions

An approximate general mathematical form of weight function is proposed which has simpli?ed the determination of weight functions for embedded elliptical cracks.Based on this general form,the point load weight functions are derived for embedded elliptical cracks in an in?nite body and in a semi-in?nite body.One reference stress intensity factor solution is used to derive these weight functions.It is demonstrated that this method gives very accurate weight functions for the wide range of geometric con?gurations for embedded elliptical cracks (for the aspect ratio range 0.16a /c 61.0).The derived weight functions are suitable for calculating stress intensity factors for embedded elliptical cracks under any complex two-dimensional stress distributions.They are particularly useful for the fatigue crack growth analysis of planer embedded cracks subjected to ?uctuating non-linear stress ?elds resulting from surface treatment (shot peening),stress concentration or welding (residual stress).

The new weight function form can also serve as the foundation for the further development of weight functions for two-dimen-sional surface cracks,corner cracks and other part-through cracks in engineering structures.Acknowledgements

The authors gratefully acknowledge the ?nancial support from the Natural Sciences and Engineering Research Council (NSERC)of Canada.Part of the work was conducted during one of the authors’(X.W.)sabbatical visit to the Department of Mechanical Engineering,University of Waterloo;he gratefully acknowledges the support from both the host and home organizations.Appendix A1

Weight function parameters M (/,a )for an embedded elliptical crack in an in?nite body presented in Table 1have been ?tted using empirical formulas with the maximum differences generally within 3%or better.

Applicable range:062/

61,and 0:16a <0:8.M factor

M e/;a T?1

à0:1104à9:0633e2/p T2

à3:9542ea T

1:5

h i eA1T

a ?

a c

Comparison of the curve ?tting predictions and the numerical data is shown in Fig.9.Appendix A2

Weight function parameters M (/,a ,d /a )for an embedded elliptical crack in a semi-in?nite body presented in Table 3at three points,point A (corresponds to /=àp /2),B (/=0)and C (/=p /2),have been ?tted into empirical formulas with the maximum differ-ences generally within 4%or better.

Applicable range:0:256d

60:625,and 0:26a <1:0.

M factor at Point A (/=àp /2)

M A

a ;d

?E A tF A a tG A d tH A a 2

tI A

d 2tJ A a d eA2T

E A ?1:8495

F A ?à0:7826

G A ?à3:9999

H A ?à0:07223

I A ?2:6881

J A ?0:7203

a ?

a c

M factor at Point B (/=p /2)

M B ea ;d T?E B tF B a à1tG B d tH B a à2

tI B

d 2

tJ B a à1

a eA3T

E B ?à0:2257

F B ?0:2606

G B ?à0:2548

H B ?à0:03145

I B ?0:1671

J B ?à0:03906

a ?

a M factor at Point C (/=0)

M C ea ;d a T?E C tF C a à1tG C d a tH C a à2tI C

d a

tJ C a à1

a eA4T

E C ?0:07905

F C ?à0:02209

G C ?à0:1046

H C ?à0:06776

I C ?à0:02864

J C ?à0:05068

a ?

a c

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