NASA Technical Memorandum 85793

j' .

STRESS-INTENSITY FACTOR EQUATIONS FOR

CRACKS IN THREE-DIMENSIONAL FINITE BODIES

SUBJECTED TO TENSION AND BENDING LOADS

J. C. Newman, Jr. and I. S. Raju

April 1984

NASA

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23665
I I
STRESS-INTENSITY FACTOR EQUATIONS FOR CRACKS IN THREE-DIMENSIONAL

FINITE BODIES SUBJECTED TO TENSION AND BENDING LOADS

J. C. Newman, Jr. 1 and I. S. Raju 2

SUMMARY

Stress-intenslty factor equations are presented for an embedded elliptical crack, a semi-elllptlcal surface crack, a quarter-elllptlcal corner crack,

a seml-elliptlcal surface crack along the bore of a circular hole, and a

quarter-elllptlcal corner crack at the edge of a circular hole in finite

plates. The plates were subjected to either remote tension or bending loads.

The stress-lntensity factors used to develop these equations were obtained

from previous three-dimenslonal flnlte-element analyses of these crack configurations. The equations give stress-lntenslty factors as a function of

parametric angle, crack depth, crack length, plate thickness, and, where

applicable, hole radius. The ratio of crack depth to plate thickness ranged

from 0 to I, the ratio of crack depth to crack length ranged from 0.2 to 2,

and the ratio of hole radius to plate thickness ranged from 0.5 to 2. The

effects of plate width on stress-intensity variations along the crack front

were also included, but were either based on solutions of similar configurations or based on engineering estimates.

INTRODUCTION

In aircraft structures, fatigue failures usually occur from the initiation

and propagation of cracks from notches or defects in the material that are

either embedded, on the surface, or at a corner. These cracks propagate with

elliptic or near-elllptic crack fronts. To predict crack-propagation life and

iSenior Scientist, National Aeronautics and Space Administration, Langley

2Research Center, IIampton, Virginia 23665

Vigyan Research Associates, Hampton, Virginia
fracture strength, accurate stress-intensity factor solutions are needed for

these crack configurations. But, because of the complexities of such problems,

exact solutions are not available. Instead, investigators have had to use

approximate analytical methods, experimental methods, or engineering estimates

to obtain the stress-intensity factors.

Very few exact solutions for three-dimensional cracked bodies are

available in the literature. One of these, an elliptical crack in an infinite

solid subjected to uniform tension, was derived by Irwin [I] using an exact

stress analysis by Green and Sneddon[2]. Kassir and Sih [3], Shah and

Kobayashi [4], and ViJayakumar and Atluri [5] have obtained closed-form

solutions for an elliptical crack in an infinite solid subjected to nonuniform loadings.

For finite bodies, all solutions have required approximate analytical

methods. For a semi-circular surface crack in a semi-infinite solid and a

semi-elliptical surface crack in a plate of finite thickness, Smith, Emery,

and Kobayashi [6], and Kobayashi [7], respectively, used the alternating

method to obtain stress-intensity factors along the crack front. Raju and

Newman[8,9] used the finite-element method; Heliot, Labbens, and PellissierTanon [i0] used the boundary-integral equation method; and Nishioka and

Atluri [Ii] used the flnite-element alternating method to obtain the same

information. For a quarter-elliptic corner crack in a plate, Tracey [12] and

Pickard [13] used the finite-element method; Kobayashi and Enetanya [14] used

the alternating method. Shah [15] estimated the stress-intensity factors for

a surface crack emanating from a circular hole. For a single corner crack

emanating from a circular hole in a plate, Smith and Kullgren [16] used a

finite-element-alternating method to obtain the stress-lntensity factors.

Hechmerand Bloom [17] and Raju and Newman[18] used the finite-element method
for two symmetric corner cracks emanating from a hole in a plate. Most of

these results were for limited ranges of parameters and were presented in the

form of curves or tables. For ease of computation, however, results expressed

in the form of equations are preferable.

The present paper presents equations for the stress-lntenslty factors for

a wide variety of three-dlmenslonal crack configurations subjected to either

uniform remote tension or bending loads as a function of parametric angle,

crack depth, crack length, plate thickness, and hole radius (where

applicable); for example, see Figure I. The equations for uniform remote

tension were obtained from Reference 19. The tension equations, however, are

repeated here for completeness and because the correction factors for remote

bending are modifications of the tension equations. The crack configurations

considered, shown in Figure 2, include: an embedded elliptical crack, a semielliptical surface crack, a quarter-elllptical corner crack, a seml-elllptlcal

surface crack at a circular hole, and a quarter-elliptical corner crack at a

circular hole in flnlte-thlckness plates. The equations were based on stressintensity factors obtained from three-dlmenslonal flnlte-element analyses [8,

9, 18, and 19] that cover a wide range of configuration parameters. In some

configurations, the range of the equation was extended by using stressintensity factor solutions for a through crack in a similar configuration. In

these equations, the ratio of crack depth to plate thickness (a/t) ranged from

0 to I, the ratio of crack depth to crack length (a/c) ranged from 0.2 to 2,

and the ratio of hole radius to plate thickness (r/t) ranged from 0.5 to 2.

The effects of plate width (b) on stress-lntenslty variations along the crack

front were also included, but were either based on solutions of similar configurations or based on engineering estimates.
a

b

c

F c

Fch

F e

Fj

Fs

Fsh

fw

f

gi

H c

Hch

Hj

H s

h

K

M

M i

Q

r

Sb

S t

NOMENCLATURE

depth of crack

width or half-width of cracked plate (see Fig. 2)

half-length of crack

boundary-correction factor for corner crack in a plate under tension

boundary-correction factor for corner crack at a hole in a plate under

tension

boundary-correctlon factor for embedded crack in a plate under tension

boundary-correctlon factor on stress intensity for remote tension

boundary-correction factor for surface crack in a plate under tension

boundary-correctlon factor for surface crack at a hole in a plate

under tension

flnlte-width correction factor

angular function derived from embedded elliptical crack solution

curve fitting functions defined in text

bending multiplier for corner crack in a plate

bending multiplier for corner crack at a hole in a plate

bending multiplier on stress intensity for remote bending

bending multiplier for surface crack in a plate

half-length of cracked plate

stress-lntenslty factor (mode I)

applied bending moment

curve fitting functions defined in text (i = i, 2, or 3)

shape factor for elliptical crack

radius of hole

remote bending stress on outer fiber, 3M/bt 2

remote uniform tension stress

thickness or one-half plate thickness (see Fig. 2)
function defined in text

Polsson's ratio (v = 0.3)

parametric angle of ellipse, deg

STRESS-INTENSITY EQUATIONS

The stress-lntenslty factor, K, at any point along the crack front in a

flnlte-thlckness plate, such as that shown in Figure I, was taken to be

K = (St + HjSB)_ Fj (la)

where

and

a 4

(ib)

Hj - H I + (H 2 - HI) sin p _ (Ic)

The function Q is the shape factor for an ellipse and is given by the square

of the complete elliptic integral of the second kind [2]. The boundarycorrection factor, Fj, accounts for the influence of various boundaries and

is a function of crack depth, crack length, hole radius (where applicable),

plate thickness, plate width, and the parametric angle of the ellipse. The

product HjFj is the corresponding bending correction. The subscript j

denotes the crack configuration: j = c is for a corner crack in a plate,

J ffie is for an embedded crack in a plate, j = s is for a surface crack in

a plate, j ffish is for a surface crack at a hole in a plate, and

for a corner crack at a hole in a plate. Functions MI, M2, M3,

and p are defined for each appropriate configuration and loading.

series containing M i

point. The function

J = ch is

H1 , H2 ,

The

is the boundary-correctlon factor at the maximum depth

f_ is an angular function derived from the solution for
an elliptical crack in an infinite solid. This function accounts for most of

the angular variation in stress-intensity factors. The function fw is a

finlte-width correction factor. Function g denotes a product of functions,

such as

and H2

zero and

define the parametric angle, $, for a/c

unity.

Very useful empirical expressions for

(see Ref. 9). The expressions are

glg2...gn, that are used to flne-tune the equations. Functions H I

are bending multipliers obtained from bending results at $ equal

_/2, respectively. Figure 3 shows the coordinate system used to

less than and a/c greater than

Q have been developed by Rawe

1.65

Q = i + 1.464(a) for a _ I

C

(2a)

1.65

Q ffiI + 1.464(c) for a > i

C

(2b)

The maximum error in the stress-intensity factor caused by using these

approximate equations for Q is about 0.13 percent for all values of a/c.

(Rawe's original equation was written in terms of a/2c).

In the following sections, the stress-lntensity factor equations for

embedded elliptical cracks, seml-elllptlcal surface cracks, quarter-elllptlcal

corner cracks, seml-elllptlcal surface cracks at a hole, and quarterelliptical corner cracks at a hole in finite plates (see Fig. 2) subjected to

either remote tension or bending loads are presented. The particular functions chosen were obtained from curve fitting to flnlte-element results [8, 9,

18, and 19] by using polynomials in terms of a/c, a/t, and angular functions of $. For cracks emanating from holes, polynomial equations in terms

of c/r and $ were also used. Typical results will be presented for

6
a/c = 0.2, 0.5, I, and 2 with a/t

range of applicability of $, a/t,

proposed equations.

varying from 0 to I. Table 1 gives the

a/c, r/t, and (r + c)/b for the

Embedded Elliptical Crack

The stress-lntensity factor equation for an embedded elliptical crack in

a finite plate, Figure 2(a), subjected to tension was obtained by fitting

equation (I) to flnlte-element results in Reference 19. The results of

Irwin [I] were used to account for the limiting behavior as a/c approaches

zero or infinity. The equation is

(3)

for 0 < a/c < =, c/b < 0.5, and -= _ $ < _ provided that a/t satisfies:

-_t < 1.2s,c+r-_0.6)for0 __-c _0.2

a< I for 0.2 C a- < ®

t c

(4)

The function Fe accounts for the influence of crack shape (a/c), crack size

(a/t), finite width (c/b), and angular location ($), and was chosen as

Fe - I + "2C_)2+ "3_)g f,fw

(s)

The term in brackets gives the boundary-correctlon factors at $ = _/2 (where

g = f$ = i). The function f$ was taken from the exact solution for an

embedded elliptical crack in an infinite solid [I] and fw is a finite-wldth

correction factor. The function g is a flne-tunlng curve-fitting function.
For a/c < 1:

H1 - I (6)

0.05

M2 ffi a 3/2 (7)

o.11+

M3., 0.29 3/2 (8)

0.23 + (_)

4 1/2

(_) (2.6- 2 _)

g = 1 - I cos )] (9)

1+

and

_1/4

= a 2 2

f$ C ) cos $ + sln 2 (I0)

(Note that eq. (9) is slightly different, and is believed to be more accurate,

than that given in Ref. 19.) The finlte-width correction, fw' from

Reference 9 was

fw ec _c (Ii)

for c/b < 0.5. (Note that for the embedded crack,

of the full plate thickness.)

t is defined as one-half

For a/c > i:

M I = _ (I2)
and

f$ = sin 2 $+ cos (13)

The functions M2, M3, g, and fw are the same as equations (7), (8), (9),

and (Ii), respectively.

Figure 4 shows some typical boundary-correctlon factors for various crack

shapes (a/c = 0.2, 0.5, I, and 2) with a/t equal to O, 0.5, 0.75, and I.

The correction factor, Fe, is plotted against the parameter angle, $. At

= O, the point on the crack front that is located at the center of the

plate, the influence of plate thickness is nmch less than at $ ffi_/2, the

point that is located closest to the plate surface. The results shown for

a/t = 0 are the exact solutions for an elliptical crack in an infinite solid

[I]. For a/t < 0.8, the results from the equation are within about 3 percent

of the flnlte-element results. (Herein, "percent" error is defined as the

difference between the equation and the flnlte-element results normalized by

the maximum value for that particular case. This definition is necessary

because, in some cases, the stress-lntenslty factor ranges from positive to

negative along the crack front.) For a/t > 0.8, the accuracy of equation (3)

has not been established. But its use in that range appears to be supported

by estimates based on an embedded crack approaching a through crack (see

Ref. 19).

Bending equations were not developed for the embedded elliptical crack.

Semi-Elllptlcal Surface Crack

The equations for the stress-lntenslty factors for a semi-elliptlcal

surface crack in a finite plate, Figure 2(b), subjected to remote tension and

bending loads were obtained from Reference 9. The tension and bending

9
equations were previously fitted to finite-element results from RaJu and

Newman[8] for a/c values less than or equal to unity. Equations for

tension and bending loads for a/c greater than unity were developed

herein. The results of Gross and Srawley [20] for a single-edge crack

were used to account for the limiting behavior as a/c aproaches zero.

The equation is

K = (S t + HsS b) _Fs(_, at, cb, $) (14)

for 0 < a/c _ 2, c/b < 0.5, and 0 < ¢ < =, again, provided that a/t

satisfies equation (4). The function Fs was chosen to be

Fs - I + "2(_) + "3(_) g f¢fw

(15)

For a/c _ i:

_1 " 1.13- 009C_) (16)

M2 = -0.54 + 0.89 (17)

0.2 +_a

c

24

M3 u 0.5 I + 14(1 - a

0.65 + a -c) (18)

c

g - 1+ [o.1+ 0.35(Fal ] (1 - sin ¢)2 (19)

and f¢ is given by equation (I0). The flnlte-width correction, fw, is

again given by equation (ii). Equations (15) through (19) were taken from

Reference 9. (The large power in eq. (18) was needed to fit the behavior as

a/c approaches zero.)

I0
The bending multiplier, Hi, in equation (I) has the form

Hj = HI + (H2 - HI) sin p $ (2O)

where HI, H2, and p are defined for each crack configuration considered.

For the surface crack (j = s),

p = 0.2 + a + 0.6 a c F (21)

a a a

H1 = 1 - 0.34_- 0.11_(_) (22)

and

2

H2 " I + G21It ) + G22(t ) (23)

In this equation for H2,

G21 = -1.22 - 0.12 _ (24)

c

0.75 I .5

G22 = 0.55 - 1.05(a) + 0-47(a) (25)

Equations (21) through (25) were taken from Reference 9.

For a/c > I:

(26)

4

M2 = 0.2(c) (27)

4

M3 " -0.11(c) (28)

11
g ffii + .I + 0.35(a)(_) (I - sin $)2 (29)

and f$ and fw are given by equations (13) and (Ii), respectively,

The bending multiplier for a/c > I is also given by equation (20) where

p = 0.2 + C + 0.6 a (30)

a t

2

H 1 = i + GII _a,+ GI2 (31)

2

H2 ,= i + G21 a_ + G22 (._.) (32)

Gll ffi -0.04 - 0.41 _a (33)

0.75 i .5

GI2 = 0.55- 1.93(c) + 1.38(c) (34)

G21 ffi -2.11 + 0.77 _a (35)

and

0.75 I .5

G22 = 0.55 - 0.72(c) + 0.14(c) (36)

Figures 5 and 6 show some typical boundary-correction factors for various

surface crack shapes (a/c ffi0.2, 0.5, I, and 2) with a/t equal to O, 0.5,

and i for tension and bending, respectively. For all combinations of parameters investigated and a/t < 0.8, equation (14) was within *5 percent of the

flnlte-element results (0.2 < a/c < 2) and the single-edge crack solution

(a/c = 0). For a/t > 0.8, the accuracy of equation (14) has not been

established. However, its use in that range appears to be supported by

estimates based on a surface crack approaching a through crack.

12
The use of negative stress-lntenslty factors in the case of bending are

applicable only when there is sufficient tension to keep the crack surfaces

open; that is, the total stress-intensity factor due to combined tension and

bending must be positive.

Quarter-Elliptical Corner Crack

The stress-lntensity factor equations for a quarter-elllptical corner

crack in a finite plate, Figure 2(c), subjected to tension and bending loads

were obtained by fitting equation (i) to the flnlte-element results in Reference 19 for tension and the results in Table I for bending. The equation is

K ffi(St + HcSb) _Fc la, at' cb' _) (37)

for 0.2 < a/c < 2,

F c was chosen as

a/t < I, and 0 < _ < _/2 for The function

Fc = I + M2(t ) + M3(_) gl g2 f_ fw

(38)

For a/c < i:

M1 ffi 1.08 - 0.03(_-) (39)

M2 = -0.44 +

i .06

0.3 +_a

c

(40)

15

"3:-05+ 0 14.8(1-a) (41)

gl ffiI + E-08 + 0.4(_)_(i - sin ¢)3 (42)

13
g2 = I + .08 + 0.15(_ ) (I - cos $)3 (43)

and f$ is given by equation (i0). The flnite-width correction, fw' was

estimated herein by using the single-edge crack tension solution given in

Reference 21 (divided by 1.12) and was

f ffi 1 - 0.2%.+ 9.4k 2 - 19.4%. 3 + 27.1k 4

9/

(44)

where %. = bc_" (The width correction from Ref. 21 was divided by

1.12

because the front-face correction was already included in eq. (38).)

Equation (44) is restricted to c/b < 0.5.

As a/t approaches unity, with a/c = i and $ = O, the stressintensity factor equation (eq. (37)) for tension reduces to

K ffi St_-_ l.llf w

(45)

Equation (45) is within about I percent of the accepted solution [21] for

c/b < 0.6.

The bending multiplier,

Functions p, HI, H2, and

respectively, for a/c _ I.

Hc, has the form given by equation (20).

G21 are given by equations (21)-(24),

The function G22 is

0.75 1.5

G22 = 0.64 - 1.05(a) + 0.47(a) (46)

For a/c > i:

M1 : _ (1.08 - 0.03 ca)

(47)

2

"2 : 0 3751 ) (48)

14
2

M3 = -0.25(c) (49)

gl ffii + .08 + 0.4(_) (I - sin $)3 (50)

g2 ffi I + .08 + 0.15(-_ (I -

3

cos +) (51)

and f$ is given by equation (13). The flnlte-width correction is, again,

given by equation (44).

The bendlng-correction factor Hc is, again, given by equation (20)

where p, HI, H2, GII , GI2 , and G21 are given by equations (30)-(35),

respectively. The function G22 is given by

0.75 i .5

G22 = 0.64 - 0.72(c) + 0.14(c) (52)

Figures 7 and 8 show some typical boundary-correctlon factors for corner

cracks in plates for various crack shapes (a/c ffi0.2, 0.5, I, and 2) with

a/t varying from 0 to I for tension and bending, respectively. At a/t = 0,

the results for tension and bending are identical. As expected, for tension

the effects of a/t are much larger at lower values of a/c. Again, the use

of negative stress-lntenslty factors in this case of bending are applicable

only when there is sufficient tension to keep the crack surfaces open (stressintensity factor due to combined tension and bending must be positive).

Semi-Elliptical Surface Crack at Hole

Two symmetric surface cracks.- The stress-intensity factor equation for

two symmetric semi-elliptlcal surface cracks located along the bore of a hole

in a finite plate, Figure 2(d), subjected to tension was obtained by fitting

equation (i) to finite-element results [19]. The equation is

15
K = St _ Fsh(ca' at' rt' rb' cb' $) (53)

for 0.2 < a/c < 2, a/t < i, 0.5 < r/t < 2, (r + c)/b < 0.5, and

-_/2 < $ < _/2. (Note that here t is defined as one-half of the full plate

thickness.) The function Fsh was chosen as

Fsh = I + M2(t ) + M3(t ) gl g2 g3 f$ fw (54)

For a/c < I:

M I = i (55)

0.05 (56)

M2 = 3/2

O.ll +

M3 ffi 0.29 3/2 (57)

0.23 + (a)

gl= i -

4 a 1/2

(t) (2.6 - 2 _)

cos ¢ (58)

I + 0.358k + 1.425k 2 - 1-578k 3 + 2.156k 4

g2 =

i + O.08X 2

(59)

X = I (60)

I + --c cos(0.9 $)

r

i0

g3 :" I + 0.1(1 - COS $)2(1 -_) (61)

(Note that eq. (58) is slightly different, and is believed to be more accurate,

than that given in Ref. 19.) The function f$ is given by equation (i0).

16
The finite-width correction, fw' was taken as

fw = (s ec(Tb)sec _r [ (b_(2r - c)+ +no)_>I/2 2no

where n = I

(62)

is for a single crack and n = 2 is for two-symmetrlc cracks.

This equation was chosen to account for the effects of width on stress concentration at the hole [22] and for crack eccentricity [21]. For a/c > I:

MI =_ (63)

The functions M2, M3, gl, g2, g3, and % are given by equations (56)

through (61), and the functions f$ and fw are given by equations (13) and

(62), respectively.

Estimates for a single-surface crack.- The stress-lntenslty factors for a

slngle-surface crack located along the bore of a hole were estimated from the

present results for two symmetric surface cracks by using a conversion factor

developed by Shah [15]. The relationship between one- and two-surface cracks

was given by

!ac

4 + 2t---_

mm

(K) one + a__c_c

crack tr

(K) (64)

two

cracks

where K for two cracks must be evaluated for an infinite plate (fw = I) and

then the finlte-wldth correction for one crack must be applied. Shah had

assumed that the conversion factor was constant for all locations along the

crack front, that is, independent of the parametric angle.

Figure 9 shows some typical boundary-correctlon factors for a single

surface crack at a hole for various crack shapes (a/c = 0.2, 0.5, I, and 2)

with a/t varying from 0 to I. These results were in good agreement with

17
boundary-correctlon factors estimated by Shah [15]. The agreementwas

generally within about 2 percent except where the crack intersects the free

surface (25/_ = I). Here the equation gave results that were 2 to 5 percent

higher than those estimated by Shah.

Stress-intensity factor equations for bending were not developed for a

surface crack located at the center of a hole.

Quarter-Elliptical Corner Crack at a Hole

Two symmetric corner cracks.- The stress-lntensity factor equations for

two symmetric quarter-elliptical corner cracks at a hole in a finite plate,

Figure 2(e), subjected to remote tension and bending loads were obtained by

fitting to finlte-element results in Reference 18. The equation is

/---/-

K = (S t + HchSb) _Q Fch (ac' t' t' b' b' $)

a r r c

(65)

for 0.2 < a/c < 2, a/t < i, 0.5 < r/t < 2, (r + c)/b < 0.5, and

0 < $ < x/2. The function Fch was chosen as

Fch = I + M2(_) 2 + M3(-t) gl g2 g3 g4 f

f

¢ w

(66)

For a/c < i:

M I = 1.13 - 0.09(a) (67)

M 2 = -0.54 +

0.89

a

0.2 +--

C

(68)

M 3 = 0.5

0.65 + a

C

24

a

+ 14(1 - c) (69)

18
where

gl = 1 + _.I + 0.351t)_(I- sln _)2

1+ 0358 ÷ 1425 2 1578 3÷ 2156#

g2 =

i + 0.13k 2

(70)

(71)

= 0.85

functions

and

I

I + _ coB(p@)

r

for tension and = 0.85 - 0.25(a/t) I/4 for bending. The

g3 and g4 are given by

g3 = II + 0.04 a-c ) Ii + 0.i(i - cos @)_I 0 .85 + 0.15[_)a i/_

(72)

(73)

a a a

g4 = I - 0.7(i - _)(c- 0.2)(i - c) (74)

Functions f@ and fw are given by equations (I0) and (62), respectively.

The bending multiplier, Hch , is given by equation (20) for a/c _ I.

The terms p, HI, and H2 are given by

P = 0.I + 1.3 at + i.i a_c 0°7 alt )

2 3

H I 1 + GII _ + GI2 GI3

(75)

(76)

and

2 3

H 2 = 1 + G21 _-+a G22 (t) + G23 [t ) (77)

where

2

GII = -0.43 - 0.74 ca - 0.84 [a) (78)

2

GI2 = 1.25 - 1.19 a + 4.39_a]_cj c

(79)

19
2

GI3 _ -1.94 + 4.22 a_ 5.51(a) (80)

c

2

G21 = -1.5 - 0.04 a_c 1"73(a) (81)

2

G22 = 1.71 - 3.17 ac + 6"84(a) (82)

2

G23 = -1.28 + 2.71 a_ 5.22(a) (83) c

For a/c > I:

(84)

4

M 2 --0 (85)

4

= -0.11 [c--_ (86) M3 _a _

gl : i + _.i + 0.35(_)(-_)2](I- sln _)2 (87)

Functions g2 and k are given by equations (71) and (72). Function g3 is

given by

g3 = [I.13 - 0.09 c) + 0.i(I - cos _)i[ 0 .85 + 0.15[_)a 1/41 (88)

and g4 = I. The functions f_ and fw

and (62), respectively.

Again, the bendlng-correction factor,

The function p is given by equation (30) for a/c > I.

given by equations (75) and (76) where

are, again, given by equations (13)

Hch , is given by equation (20).

The H-functions are

20
GII = -2.07 + 0.06 ! (89)

a

GI2 = 4.35 + 0.16 _ (90)

a

GI3 = -2.93 - 0.3 _ (91)

a

G21 = -3.64 + 0.37 _ (92)

a

G22 = 5°87 - 0.49 _ (93)

a

G23 = -4.32 + 0.53 ! (94)

a

Estimates for a single-corner crack.- The stress-intenslty factors for a

slngle-corner crack at a hole were estimated from the present results for twosymmetric corner cracks by using the Shah-conversion factor (eq. (64)). Raju

and Newman [18] have evaluated the use of the conversion factor for some

corner-crack-at-a-hole configurations. The stress-intenslty factor obtained

using the conversion factor were in good agreement with the results from Smith

and Kullgren [16] for a slngle-corner crack at a hole.

Figures I0 and II show some typical boundary-correction factors for a

single corner crack at a hole for various a/c and a/t ratios for tension

and bending, respectively. Again, the use of negative stress-intensity

factors in the case of bending are applicable only when there is sufficient

tension to make the total stress-lntensity factor, due to combined tension and

bending, positive.

21
CONCLUDING REMARKS

Stress-lntenslty factors from three-dimenslonal flnlte-element analyses

were used to develop stress-lntenslty factor equations for a wide variety of

crack configurations subjected to either remote uniform tension or bending

loads. The following configurations were included: an embedded elliptical

crack, a seml-elllptlcal surface crack, a quarter-elliptlcal corner crack, a

seml-elllptlcal surface crack along the bore of a hole, and a quarterelliptical corner crack at the edge of a hole in finite plates. The equations

cover a wide range of configuration parameters. The ratio of crack depth to

plate thickness (a/t) ranged from 0 to I, the ratio of crack depth to crack

length (a/c) ranged from 0.2 to 2, and the ratio of hole radius to plate

thickness (r/t) ranged from 0.5 to 2 (where applicable). The effects of plate

width (b) on stress-lntenslty variations along the crack front were also

included, but were based on engineering estimates.

For all configurations for which ratios of crack depth to plate thickness

do not exceed 0°8, the equations are generally within 5 percent of the finiteelement results, except where the crack front intersects a free surface. Here

the proposed equations give higher stress-lntenslty factors than the finiteelement results, but these higher values probably represent the limiting

behavior as the mesh is refined near the free surface. For ratios greater

than 0°8, no solutions are available for direct comparison; however, the

equations appear reasonable on the basis of engineering estimates.

The stress-lntenslty factor equations presented herein should be useful

for correlating and predicting fatigue-crack-growth rates as well as in

computing fracture toughness and fracture loads for these types of crack

configurations.

22
REFERENCES

[I] Irwin, G. R.: The Crack Extension Force for a Part-Through Crack in a

Plate, ASME,J. Appl. Mechs., Vol. 29, No. 4, 1962, pp. 651-654.

[2] Green, A. E.; and Sneddon, I. N.: The Distribution of Stress in the

Neighborhood of a Flat Elliptical Crack in an Elastic Solid, Proc.

Cambridge Phil. Soc., Vol. 47, 1950, pp. 159-164.

[3] Kassir, M. K.; and Sih, G. C.: Three-Dimensional Stress Distribution

Around an Elliptical Crack Under Arbitrary Loadlngs, Journal of Applied

Mechanics, Vol. 88, 1966, pp. 601-611.

[4] Shah, R. C.; and Kobayashi, A. S.: Stress-Intensity Factor for an

Elliptical Crack Under Arbitrary Normal Loading, Engineering Fracture

Mechanics, Vol. 3, 1971, pp. 71-96.

[5] Vijayakumar, Ko; and Atluri, S. N.: An EmbeddedElliptical Flow in an

Infinite Solid, Subject to Arbitrary Crack-Face Tractions, Journal of

Applied Mechanics, Vol. 48, 1981, pp. 88-96.

[6] Smith, F. W.; Emery, A. F.; and Kobayashi, A. S.: Stress Intensity

Factors for Semi-Circular Cracks, Part 2 - Semi-Infinite Solid, J. Appl.

Mechs., Vol. 34, No. 4, Trans. ASME,Vol. 89, Series E, Dec. 1967,

pp. 953-959.

[7] Kobayashi, A. S.: Crack-Openlng Displacement in a Surface Flawed Plate

Subjected to Tension or Plate Bending, Proc. Second IntJ_Conf. on

Mechanical Behavior of Materials, ASM, 1976, pp. 1073-1077.

[8] RaJu, I. S.; and Newman, J. C., Jr.: Stress-Intenslty Factors for a Wide

Range of Semi-Elliptical Surface Cracks in Finite-Thickness Plates,

Engineering Fracture Mechanics J., Vol. ii, No. 4, 1979, pp. 817-829.

[9] Newman, J. C., Jr._ and RaJu, I. S.: Analyses of Surface Cracks in

Finite Plates Under Tension or Bending Loads, NASA TP-1578, Dec. 1979.

23
[I0]

[ii]

[[2]

[Z3]

[14]

[15]

[16]

[_.7]

[181

Heliot, J°; Labbens, R.; and Pellissier-Tanon, A.: Benchmark Problem

No. I - Semi-Elliptical Surface Crack, Int. J. of Fracture, Vol. 15,

Noo 6, Dec. 1979, pp. RI97-R202.

Nishioka, T.; and Atluri, S. N.: Analytical Solution for Embedded

Elliptical Cracks, and Finite Element-Alternating Method for Elliptical

Surface Cracks, Subjected to Arbitrary Loadings, Engineering Fracture

Mechanics, Vol. 17, 1983, pp. 247-268.

Tracey, D. M°: 3D Elastic Singularity Element for Evaluation of K

Along an Arbitrary Crack Front, Int. J. of Fracture, Vol. 9, 1973,

pp. 340-343.

Pickard, A° Co: Stress Intensity Factors for Cracks with Circular and

Elliptic Crack Fronts-Determined by 3D Finite Element Methods,

PNR-90035, Rolls-Royce Limited, May 1980.

Kobayashi, A° S.; and Enetanya, A. N.: Stress Intensity Factor of a

Corner Crack, Mechanics of Crack Growth, ASTM STP-590, American Society

for Testing and Materials, 1976, pp. 477-495.

Shah, R. C.: Stress Intensity Factors for Through and Part-Through

Cracks Originating at Fastener Holes, Mechanics of Crack Growth, ASTM

STP-590, American Society for Testing and Materials, 1976, pp. 429-459.

Smith, F. W.; and Kullgren, T. E.: Theoretical and Experimental

Analysis of Surface Cracks Emanating from Fastener Holes,

AFFDL-TR-76-104, Air Force Flight Dynamics Laboratory, Feb. 1977.

Heckmer, J° L.; and Bloom, J. M.: Determination of Stress Intensity

Factors for the Corner-Cracked Hole Using the Isoparametric Singularity

Element, Int. J. of Fracture, Oct. 1977.

Raju, I. S.; and Newman, J. C., Jro: Stress-Intensity Factors for Two

Symmetric Corner Cracks, Fracture Mechanics, ASTM STP-677, C. W. Smith,

Ed°, American Society of Testing and Materials, 1979, pp. 411-430.

24
[t9]

[20]

[211

[22]

Newman, J. C., Jr.; and RaJu, I. S.: Stress-Intensity Factor Equations

for Cracks in Three-Dimenslonal Finite Bodies, Fracture Mechanics -

Volume I: Theory and Analysis, ASTM STP-791, J. C. Lewis and G. Sines,

Eds., American Society of Testing and Materials, 1983, pp. 1-238, 1-265.

Gross, B.; and Srawley, J. E.: stress-Intensity Factors for SingleEdge-Notch Specimens in Bending or Combined Bending and Tension by

Boundary Collocation of a Stress function, NASA TN D-2603, 1965.

Tada, H.; Paris, P. C.; and Irwin, G. R.: The Stress Analysis of Cracks

Handbook, Del Research Corporation, 1973.

Howland, R. C. J.: On the Stresses in the Neighbourhood of a Circular

Hole in a Strip Under Tension, Philos. Trans. R. Soc. London, Series A,

Vol. 229, Jan. 1930, pp. 49-86.

25
Table I. Range of applicability for stress-intensity factor equations.

Configuration Equation _ a/t a/c r/t (r + c)/b

Embedded Crack in Plate (a) (3) -_ to _ (b) 0 to <o.5(c)

Surface Crack in Plate (14) 0 to _ (b) 0 to 2 <o.5(c)

0",

Corner Crack in Plate II

(37) 0 to _ <I 0.2 to 2

_T W

Surface Crack at Hole (a'd) (53) - _ to _-

Corner Crack at Hole (d)

<i 0.2 to 2 0.5 to 2

(65) 0 to _ (e) 0.2 to 2 0.5 to 2

<o.s(c)

<0.5

<0.5

(a)Equations for bending were not developed for this case.

(b)a/t < 1.25 (a/c + 0.6) for 0 _ a/c ( 0.2 and a/t < i

(C)r = O.

(d)one or two-symmetric cracks.

(e)a/t < I for remote tension and a/t ( 0.8

for a/c > 0.2.

for remote bending.
Table .

Boundary-correction factors, FcHc, for quarter-elliptical

corner crack in a plate subjected to bending

(v= 0"3; FHc c = K/ISb_ ))"

sit

a/c 2_/_ 0.2 0.5 0.8

0.2

0.4

1.0

2.0

0 0.522 0.609 0.779

0.25 0.669 0.702 0.808

0.5 0.801 0.746 0.716

0.75 0.868 0.746 0.577

1.0 0.876 0.750 0.604

0 0.740 0.799 0.904

0.25 0.724 0.690 0.670

0.5 0.785 0.632 0.451

0.75 0.826 0.583 0.272

1.0 0.846 0.569 0.262

0 1.084 1.046 1.027

0.25 0.934 0.770 0.604

0.5 0.838 0.547 0.237

0.75 0.798 0.417 0.011

1.0 0.839 0.407 -0.032

0 0.932 0.811 0.734

0.25 0.851 0.623 0.442

0.5 0.761 0.413 0.105

0.75 0.700 0.268 -0.131

1.0 0.677 0.215 -0.206

27
St

3M

Sb =

. .-_:!::i:::i__:_::!; .:i_:_:i!i;..:_:_,.4 L

' ."': _- -,- ' ' k ,,,._"._ :' "." .':. __-_ .z_:.,,.::_::::

__..., "" _"'":' ' ' I:Y'_::_:'*" _i""_' ....

I

I

I

/ !

St

!i:i_.i:_!:ii_::,_,y

Fig. 1--Corner cracks at the edge of a hole in a

finite plate subjected to remote tension

and bending.

28
_L

a

T

_- 2b -I

:_-"___.:.:.:.U;i!_.,_.:_i_._i:;!!!!_M_:_-._::_'_i::_i._:i._ '"_*_:"-- : _-_!ti,iii!_it-2t ii!iiiili iiiii!ili!iiiiiii ,iiiiiiiiiili,!

_L

(a) Embedded crock

I_

,- 2b ,

-..;.........;-v>.- ..........,-...........-.........:-..;....:...,..:.:..:.....:'-!."_'..b'..-':;":.'-'.k:.'.: T

• .-.. •..-,%.:.-.,°. ;.<,'...,_

... ....l. .:,.:%..e..o :

..... _' ki" • : ";.'_ "'-:: *'-":.:" .1 t

__ _i":'"'_":":__L

I-_ 2c---H

(b) Surface crock

-- b. "1

t

_L

(c) Cornercrack

I- b ..,

Ta_i c "":_'ii ._.i_

"--- 2r--_l

:I

2t

:'::'-.:.'-i" k" I

(d) Surface crack at hole

IIll_':'': _i!":':? _ii!_!_iii_iiii L:'";:'::"_?"_"_...:_r._4;.1

.::':"".:::':'."'-':.'_:.:".'::I

"-!'-;:::-:i:ii'L:::'":':"."_

2b "1

.....::=..;;..;.**.::,,._. :;t.

:..'..:.:4.'.: -..._.*,..... ::..: .,":'-:: ....:

:':':":"_v_:':""'"':':":it'::" 7"!1

.,.._. • *4,_,.-. •-.,, ..- .." ...:. •

T "':::':;'*'.".".::.'.: ::-: :.:':'. t

O :.:".-_::6:.'.;::::,:C.::;-'..:..1

.-'_.:.:::-.::.:,.-..--:._: _..

----2r---! c _

(e) Corner crock at hole

Fig. 2--Embedded-, surface-, and corner-crack configurations (all cracks have

elliptical fronts).
Y Y

Go

0

T

G

1

x

T

I

_--c--_

x

(a) ale _ i (b) a/c > 1

Fig. B--Coordinate system used to define parametric angle.
"51 I j0.75

Fe 1105 0 _

I t

(a) alc= 0.2, (b) alc = 0.5.

1,5

1,0

F e

0.5

_5

I

0,5

/ o

Exact [i] alt = 1

0 I I

0 .5 1 0 .5

2_/_ 2_/_

(c) alc = i. (d) alc = 2.

I I

1

Fig. 4--Typlcal boundary-correction factors for an embedded elliptical

crack in the center of a plate subjected to remote tension

(clb: 0).

31
F S

2

1

0

2

I

(a) alc = 0,2.

I

(b) alc = 0.5.

Fs 1 0

0 I t I

0 .5 1 .5

2_/_r 2_/_r

(c) alc = 1. (d) olc = 2,

I

1

Fig. 5--Typlcal boundary-correctlon factors for a surface crack in a

plate subjected to remote tension (c/b = 0).

32
1,5"

1,0,

FsHs ,5,

0

-,5

alt = 0 . alt = 0

,5

I

(a) alc = 0,2, (b) a/c = 0,5,

FsHs

-,5

°1

0

1'5]L

t-----..., e/t = 0

1,0,'. ---- - -

,5

,5 1 0

(c) alc = i,

.5

2_/7r

(d) a/c = 2,

1

Fig. 6--Typical boundary-correction factors for a surface crack in a

plate subjected to remote bending (c/b = 0).

33
F C

2

1

0 I I

(a)a/c = 0.2.

I I

(b) olc = 0.5,

F C

2

1

0

0

0

I I I°I

.5 1 0 .5

2@/_T 2_/_

(c) olc = 1. (d) alc = 2.

!

1

Fig. 7--Typlcal boundary-correctlon factors for a corner crack in a

plate subjected to remote tension (c/b = 0).

34
1,5_

FcHc

FcHc

i0

0

-,5

alt = 0

1

I t

(a) alc = 0,2,

1,5

L alt = 0

1'0 __,5

,5,

°I

0 ,5 1

(c) alc = i.

0

a/t = 0

i

(b) alc = 0,5.

i

,5

2W/r

(d) alc = 2,

1

Fig. 8--Typical boundary-correctlon factors for a corner crack in a

plate subjected to remote bending (c/b = 0).

35
Fsh

3

2

1

0

.2

5

1

(a) GIc = 0,2, (b) alc = 0.5.

Fsh

3

2

1

0

a/t = 0 __

,2

a/t = 0

,2

5

1

0 ,5 1 0 .5 1

2(b/_ 2_/

(c) a/c = i, (d) a/c = 2,

Fig. 9--Typical boundary-correction factors for a single surface crack

at the center of a circular hole in a plate subjected to remote

tension (r/t = l; r/b = 0).

36
a/t =

2 1

Fch

1

I t i I •

alc = 0,2. (b) alc = 0.5.

T

3Lalt = 0

Fch 1

1 1

0 I 4 l _,

0 ,5 1 0 ,5 1

(c) a/c = 1, (d) a/c = 2,

Fig. lO--Typical boundary-correction factors for a single corner crack

at the edge of a circular hole in a plate subjected to remote

tension (r/t = i; r/b = 0).

37
1, Report No. 2. Government Accession No.

NASA TM-85793

4. Title and Subtitle

STRESS-INTENSITY FACTOR EQUATIONS FOR CRACKS IN TttREEDIMENSIONAL FINITE BODIES SUBJECTED TO TENSION AND

BENDING LOADS

7. Author(s)

J. C. Newman, Jr. and I. S. Raju

9. Performing Organization Name and Address

NASA Langley Research Center

Hampton, VA 23665

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, DC 20546

3. Recipient's Catalog No.

5. Report Date

April 1984

6. PerformingOrganizationCode

505-33-23-02

8. Performing Organization Report No,

10. Work Unit No,

'11. Contract or Grant No,

i3. Type of Report and Period Covered

Technical Memorandum

14, Sponsoring Agency Code

15. Supplementary Notes

16. Abstract

Stress-intensity factor equations are presented for an embedded elliptical

crack, a semi-elliptical surface crack, a quarter-elliptical corner crack, a semielliptical surface crack along the bore of a circular hole, and a quarterelliptical corner crack at the edge of a circular hole in finite plates. The

plates were subjected to either remote tension or bending loads. The stressintensity factors used to develop these equations were obtained from previous

three-dimensional finite-element analyses of these crack configurations. The

equations give stress-intensity factors as a function of parametric angle, crack

depth, crack length, plate thickness, and, where applicable, hole radius. The

ratio of crack depth to plate thickness ranged from 0 to I, the ratio of crack

depth to crack length ranged :from 0.2 to 2, and the ratio of hole radius to plate

thickness ranged from 0.5 to 2. The effects of plate width on stress-intensity

variations along the crack front were also included, but were either based on

solutions of similar configurations or based on engineering estimates.

17. Key Words (Suggested by Author(s))

Finite-element analysis

Stress-intensity factor

Surface cracks

Corner cracks

19. S_urity Cla=if. (of this report) 20. Security Classif. (of this pege)

Unclassified Unclassified

18, Distribution Statement

Unclassified - Unlimited

Subject Category 39

21. No. of Pages 22. _ice

39 A03

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