. ' 

NASA Contractor Report 177926 

INTERACTION OF PART-THROUGH 

CRACKS IN A FLAT PLATE 

B. Aksel and F. Erdogan 

LEHIGH UNIVERSITY 

Bethlehem, Pennsylvania 

Grant NGR 39-007-011 

Apr; 1 1985 

NJ\SJ\ 

National Aeronautics ana 

Space Administration 

langley Research Center 

Hampton. Virginia 23665 

NASA-CR-177926 

19850013382 

J l..'. '1 ' 

Ll!%LEY fiES[,,,,CH ClNTm 

L!2nA."?'(, rJI\SA 

/-!';',?TJN, VIRGltllA 

1111111111111 1111 1111111111 111111111111111111 

NF00690 
ABSTRACT 

Main purpose of this study is to determine the accuracy of the 

line spring model, to investigate the effect of interaction between 

two and three cracks, 

which may be useful 

and to provide extensive numerical results 

in applications. Line spring model with 

Reissner's plate theory is formulated to be used for any number and 

configurations of cracks provided that there is symmetry. This model 

is ~sed to find stress intensity factors for elliptic internal 

cracks, elliptiC edge cracks and two oppOSite elliptic edge cracks. 

Unfortunately, because of the unavailability of previous work done 

on the cases considered, only stress intensity factors for central 

elliptic crack could be compared with other methods. Despite the 

simplici ty of the line spring model, the results are found to be 

surprisingly close. 
1. INTRODUCTION 

From the viewpoint of practical applications, the analysis of a 

part through crack in a structural canponent which may locally be 

represented by a "plate" or a "shell" is certainly om of the most 

important problems in fracture mechanics. In its general form, the 

problem is a three-dimensioml crack problem in a bounded geanetry 

where the stress fields perturbed by the crack interacts very 

strongly with the surfaces of the solid. At present, even for the 

linearly elastic solids, a neat analytical treatment of the problem 

very heav ily rely on some kind of numerical techniq ue , such as 

altermting method, [5], [6], or boundary integral equation method, 

[7]; but most notably on the finite element method, [8]-[10]. The 

renewed interest in recent years in the so called "line-spring 

model" first described in [3] has been due partly to the desire of 

providing simpler and less expensive solutions to the part-through 

crack problem and partly to the fact that for certain important 

crack geanetries, the model seems to give results that have an 

acceptable degree of accuracy. 

In a plate or shell containing a part-through crack and 

subjected to membrane and bending loads, the net ligament(s) around 

the crack would generally have a constraining effect on the crack 

surface displacements and rotations. The basic idea underlying the 

"line-spring model" conSJ.sts of approxima ting the three-dimensional 

2 
t 

crack problem by a two-dimensional coupled bending-membrane problem 

through the reduction of the mt ligament stresses to the neutral 

surface of the plate or shell as a membrane load N and a bending 

moment M. In the resulting two-dimensional problem, the crack 

surface displacements are represented by a crack opening 

displacement 6 and a crack surface rotation', referred to, again, 

the neutral surface. The quanti ties N, M, & and e are assumed to be 

functions of a single var~able, namely the coordinate x1' along the 

crack in the neutral surface (Fig. 1). The pair of functions (6,8) 

or (N,M) are detennined fran the corresponding mixed boundary value 

problem for the plate or the shell having a through crack in which N . and M are treated as unknown crack surface loads. Once Nand M are 

determined, the stress intensity factors are evaluated from the twodimensional elasticity solution of a strip under the membrane force 

N and the bending moment M. 

'!he model introduced in [3] is based on the classical theory. 

However, the asymptotic stress field around the crack tip given by 

classical plate bending theory is oot consistant with the elasticity 

solutiOns, whereas a transverse shear theory (such as that of 

Reissner's) which can accanodate all stress and manent resultants on 

the crack surface seperately (i. e., three boundary conditions in 

plates, five in shells) give results which are identical to the 

asymptotic solutions obtained from the plane strain and anti-plane 

shear crack problens [2], [11]. '!be line spring model was later used 

3 
in [12], [13] to treat the longitudinal part-through crack problem 

in a cylinder by using the classical shell theory and in [1], [14] 

by using transverse shear theory. Rather extensive results for 

corner cracks and for collinear surface cracks in a plate having a 

finite width are given in [15]. The similar problem of sur-face 

crack-boundary interaction in a cylindrical shell wi th a free or a 

fixed boundary is considered in [16] and [17]. 

In this study, stress intensity factors for elliptic internal 

cracks under pur-e bending and tension, and for collinear elliptic 

edge cracks under pur-e tension will be studied by using the linespring model with Reissner's plate bending theory. Interaction of 

two and three identical elliptic edge and internal creaks will also 

be _studied by using the same method. 

4 
I 

I 

t 

2. FORMULATION OF THE PROBLEM 

2. 1 Governing Integral Equations 

The problem under consideration is a surface or an internal 

crack problem for a rela tively thin-walled structural canponent 

which is solved basically as a plate or shell problem. In the usual 

nota tion u1, u2' u3 are the canponents of' the displacement vector, 

'1 and' 2 are the angles of rotation of the normal to the neutral 

surface in x 1x3 and :X:~3 planes, res}Ectively, and Nij' Mij and 

Vi (i,j:1,2) are respectively the membr~, moment and transverse 

shear resultants (Fig. 1). 

Related mixed boundary value problem for any nUlllber of cracks 

has been formulated in [15] by using Reissner's plate bending 

theory. The derivatives of the crack surface displacement and the 

'crack surface rotation on the neutral surface are chosen as the 

unknown f'unctions. Fredholm kernels k1(x,t) and k2 (x,t) in [15] 

represent the effect of finite length of the plate in x, direction. 

!bey will vamsh in this case since the plate has an inf'inite length 

in x1 direction. Integral equations are written for only half of the 

infin1 te plate (x1 > 0) and are only valid when symmetry with 

respect to the ~x3 plane eXists. D represents the normalized 

cracked region on the x1 axis, which means that if there is a crack 

in the region (b, c), D represents the region (b/a , cIa) where "a" 

5 
------- --- ---- - - - --------

is the half crack length and equals to (c-b )/2. x is also normalized 

parameter on Xl axis which is x=x1/ai' where ai the half crack 

length for the corresponding crack(i subscript is needed because 

general formulation is done for more than one crack). 

From [15], integral eqt.a tions are as follows: 

4r(X) 1 f 1 1 "o(x) 

--- - [-+ -]G1(t) dt =-- E 2. t-x t+x E 

x€ D , ( 1) 

o 

= x€ D , ( 2) 

where K2 is the modified Bessel function of the seoond kind and the 

constants R1 , R2, R3, R4 and pare def !ned in Appendix I. h is the 

thickness of the plate, 2a is the length and L(x1) is the depth of 

the crack (Fig. 1). E is the modulus of elasticity and II' is the 

Poisson's ra tio of the material. lbe unknown functions are defined 

by 

6 
( 3) 

'!he external loads 

=- ( 4) 

h 

represent uniform membrane and bending resultants applied to the 

plate away from the crack region and cr and m which are defined by 

N(x,O) 

~(x) =---

h 

6M(x,0) 

m(x) =-~­

h2 

x€D , ( 5) 

are the membrane and bending loads applied to the crack surfaces 

(Fig. 1). The integral equa tions are obtained fran the following 

mixed boundary condi tiona in x2=0 plane 

7 
N22 (x,O) = -N~2 + N(x) x ED , (6a) 

u2 (x,O) = 0 xED , (6b) 

~2(x,O) = -~2 + M(x) x ED , (7a) 

.. 2(x,O) = 0 xED, (7b) 

where the general principle of superposition is used to acoount for 

the loading ~2' and ~2 applied to the structur-e away fran the 

crack region. 

2.2 Line-Spring Model for an Edge Crack 

Edge crack is chosen for introductory purposes because of the 

simplici ty of its formulation. Mod1fica. tiona will be made as more 

canplica.ted crack geanetries are examined. 

The oonfiguration studied here is an ninf1ni ten elastic plate 

of thickness h, whJ.ch contains a surface crack of length 2a and 

depth L(x,) penetrating part through the thickness (Fig. '). For 

general purposes Lex, )can be any function wh~ch enables us to treat 

any crack geanetry. At remote distances from the crack site, the 

plate is subjected to loads eqmpollent to a uniform smple tension 

in the x2 direction and to pure bending about the x, axis. The first 

8 
assumption in the line-spring analysis i~ that the stress intensity 

factor at a point along the crack front with coordinate x1 is 

identical to the stress intensity factor for an edge cracked strip 

in plane strain (Fig. 2), subjected to an axial force and manent 

equal, respectively, to N22 (x1 , 0) and M22(x1, 0), and having a crack 

depth equal to L(x1). In [1], [3], mode I stress intensity factor 

for the plane strain problEm is given as 

( 9) 

where functions gt(s) and gb(s) are called shape functions for 

tenSion and bending respectively, and given as 

~ 

gt(s) = V'fI s L b

i s2( i-1) (10a) 

i. .. I 

t\. 

gb(s) = J'fI s I CiS ( i-1 ) i., (10b) 

The value of n is chosen according to the desired accUl'acy. lhe 

coefficients bi and ci can be found by a suitable curve fitting 

(Appendix II). Coefficients of the shape functions for a number of 

crack geanetries are given in Tables 2, 4, 6, 8, 10, 12, 14, 16, 17. 

9 
In order to obtain N(x1) and M(x1) in terms of G1 and G2, the 

energy available for fracture along the crack front is expressed in 

two different ways, namely as the crack closure energy and as the 

product of load-load point displacement. In a plate with an edge 

crack subjected to uniform tension N and uniform bending manent M 

(Fig. 2), if K is the stress intensi ty fact or given by the plane 

strain solUtion, from the crack closure energy, the energy (per unit 

width) available for fracture may be obtained as 

() 1_,2 

G = - (U-V) = _ K2 

OL E 

(11 ) 

where U is the work done by the external loads and V is the strain 

energy. 

Let 6 and e be the load line displacements corresponding to N 

and M shown in Fig. 2 and d6 and de be the changes in 6 and 8 as 

the crack length goes from L to L+dL under fixed load conditions. 

Then referring to Fig. 2, the changes in U and V may be expressed as 

dU = Nd6 + Mdt ( 12) 

dV = (112) [N(6+d6) + M(8+d8) - (112) (N6 + ~) = 

= (11 2) (Nd6 + Mdt) ( 13) 

Equa tions (12) and (13) give the energy avaJ.lable for a crack growth 

dL as follows! 

10 
d(U-V) = (112) (Nd6 + Mde) (14) 

On the other hand for constant, Nand M, and for a change of elL in 

the crack length, we have 

oe 

de =-dL • 

OL 

nius, from (14) and (15) it follows tha t 

a 

-(U-V) 

OL 

and, by using (11) we find 

( 15) 

(16) 

(17) 

From (5) and (8), we may rewrite (17) in terms of • and m as 

11 
fOllows: 

In order to solve f$ and m in terms of 6 and e or vice versa we 

introduce the so called canpllance coefficients. In a cracked strip, 

displacement and rotation are functions of both the applied stress 

and bending manent. This rela tion may be eXll"essed as 

( 19a) 

(19b) 

where (with proper normalizations) Atb=Abt by elastic reciprOcity. 

The compliance coefficients AiJ depend only on L and vanish when 

L=O. If we SUbstitute (19) into (18) and eqm te canmon coefficients 

in the quadratic forms of both sides, we get 

dAtt 2(1-,2)h 

-=----g~ 

dL E 

(20a) 

12 
- =-= ----gtgb' 

dL dL E 

dAbb 2( 1-p2)h 

------g~ dL E 

By defining 

L 

cz1j = h-1 J g1gj dL 

o 

(i,j = b,t) , 

and knowing that Aij=O at L=O, we may write 

2h2(~_p2) 

Att = ---- a tt 

E 

2h2(1_p2) 

Abb = abb 

E 

13 

(20b) 

(200) 

-(21) 

(22a) 

(22b) 

(220) 
Substitution of (22) into (19) gives 

2h(1-,2) 

& = (CXttCJ' + C%tbm) , (23) 

E 

12( ,_,2) 

e = 

E 

(CXbt<r + cxbbm) , (24) 

where CX tb=CXbt• From (3), & and 8 may be expressed in terms of the 

unknown functions G 1 and G2 as 

x 

8 = 2_2(x,+O) = 2 J G,(t) dt (25) 

-/ X 

6 = 2au2 (x,+O) = 2a J G2(t) dt • (26 ) 

-I 

If we solve (23) and (24) for <rex) and m(x) and substitute (25) and 

(26) for & and 8, we can determine cr(x) and m(x) in terms of the 

unknown functions G1 and G2 as follows: 

cr(x) 

x 

= E [4tf tt(x) J G2 (t) 

-I 

~ 

dt + "Vtb(x) J G1(t) 

-I 

14 

dt] , (27) 
m(x) 

where 

x 

= 6E ['I' bt(x) f G2(t) 

-I 

1 Cl bt :- - 6h( 1_,2) 4 

1 Cl tt 

*Ybb = --~-

36( 1_,2) 4 

)C 

dt + 'I'bb(x) f GI(t) dtl , 

-I 

(28) 

(29a) 

(29b) 

(29c) 

(29d) 

(2ge) 

We should keep in mind that ClC ij is a function of L(x)/h, which means 

a ij is al so a function of x. 

15 
For a single crack shown in Fig. 1, the normalized cracked 

region on x1 axis (called D in (1) and (2» is defined as (0,1). (1) 

and (2) then become 

I 

er(x) - ~ f [~+ 1 er o - ]G1 (t) dt =-, o < x < 1 , (30) 

E 21r t-x t+x E 

0 

I 

m(x) 

Rl Jh 1 1 1 1 

-- -+-- - R3 [ + (t+x)3 ] 6E t-x t+x (t-x)3 

0 

0<x<1. (31) 

6E 

For convenience in the numerical solution of integral eqm tions we 

need to convert the integrals calculated over (0,1) to (-1,1). 

Because of that the geanetry and loading conditions are symmetric 

with respect to the x2x3 plane, u2 (x,+0) and' 2(x,+0) should also 

be symmetric with respect to the same plane. In mathemathical form 

(32a) 

16 
Observing that derivative of' a symmetric function is antisymmetric, 

we may write 

(33a) 

(33b) 

By applying the above properties, integrand of (30) can be rewritten 

as 

(34) 

By a change of variable t1 = -t and from (33a), we can write 

17 
Substituting (35) in (34) 

1 

f 1 1 [-+ -] G1(t) 

t-x t+x 

o 

-I 

= f ~ [-G;(t)] (-dt) 

-t+x 

o 

o 

= f _1_ G1 (t) dt • 

t-x 

-I 

1 

dt =f ~G1 (t) 

t-x 

-I 

dt • 

(35) 

(36) 

At'ter applying the same procedure to (31) and substituting (27) and 

(28) for CS' (x) and m(x), integral aqua tiona take the following final 

form 

)( 

'l'bb(x) f G, (t) dt -

-I 

x 

+ R4~ K2 (P/t-x l) J G1(t) dt + "'-bt(x) J G2 (t) dt 

t-x 

-I 

=- -1 < x < 1 

6E 

18 

(37a) 
x x 

... tb(xl I G, (tl dt + ... tt(xl I G2(tl dt 

-I 1 -I 

_ 2-f~(t)dt __ ero 

-1 < x < 1 • 

211' t-x E 

(37b) 

-I 

From (3) it follows that the unmown functions G, and G2 must 

satisfy the single valuedness conditions given by 

1 

IG1(t)dt=o, 

-I 

(38) 

After solving (37) and (38) by using Gauss-Chebyshev closed type 

in~egra tion formula which is defined in Appendix III, a backward 

procedure is applied to find the stress intensity factors along the 

crack front. Unmown functions are integrated (Appendix IV) from -1 

to the value of x at which the stress intensity factor is desired. 

!ben those integrals are substituted to (27) and (28) with the 

functi~ns 'fij(x), (i,j=t,b) evaluated at that particular point to 

get corresponding er and m. Once er and m are found, (8) gives the 

stress intensity factor at a specifiC point on the crack front. 

19 
2.3 Internal Cracks 

The line-spring model described in the previous section may 

easily be extended to internal cracks such as that shown in Fig. 3. 

In this case the basic integral equations remain the same and again 

are given by (1) and (2). However, there are two crack tips (shown 

as A and B in Fig. 3) which create two different stress intensity 

factors for each cross-section taken perpendicular to x1 direction. 

Previous assumptions regarding the stress intensity factors will 

also remain the same; however, they are now defined seperately, 

i. e. , 

KA = {h [4rgAt(s) + mgAb(s)] 

KB = {h [crgBt(s) + mgBb(s)] 

(39a) 

(39b) 

where gAt' gAb' gBt' gBb are the shape functions corresponding to 

crack tips A and B. Unlike the edge crack case, these shape 

functions are now functions of both L(x1 )/h and dlh where d ~s the 

distance from the center of the crack to the neutral plane. Two way 

20 
parametrization is needed in order to define the shape functions 

which are valid for any kind of internal crack. Since ell~ptic 

cracks whose major axis are parallel to the free surface are the 

main concern in the present study, parametrjza tion can be simplified 

by observing that the distance between the center of any crack 

obtained by taking cross-sections perpendicular to the major axis 

and the center line of the plate is fixed. In other words, d is 

constant for any elliptic crack whose major axis is parallel to the 

free surface. Because of this property, as in the previous case, we 

may again define the shape functions for each crack as a function of 

only one variable, which is L(x1 )/h, as 

"-

gAt = V#lfS l: b s2(i-1) 

1..1 Ai (40a) 

n. 

r:r;;. ~ i-1 (40b) gAb = #lfs cAis "., 

n. 

gBt =..;.s I~ s2(i-1) (40c) 

• I i \.. 

f\. 

gBb = V#lfS I cBisi - 1 ( 40d) 

ial 

Keep in mind that coefficients are different for each val ue of d. 

The next step is the representation of the energy available for 

fracture in terms of both stress intensity factors and product of 

load-load po~nt displacement. As L increases by dL, the energy 

21 
increment available for fracture may be expressed as 

giving 

1_,2 

d(tJ-V) = -[ Ki d(Ll2) + K~ d(Ll2) ] 

E 

(41) 

(42) 

which replaces (11). The rate of energy available for fracture as 

expressed in terms of the load line displacements and forces remains 

the same and is given by (16). If the same procedure as in the 

previous section is followed, the functions CX(ij) may be found as 

L 

CX tt = h- 1 J (git + g~t)/2 dL (43a) 

°L 

CXbb = h-1 J (gib + g§b)/2 dL (43b) 

° L 

CX tb = CX bt = h- 1 J (gAtgAb + gBtgBb)/2 dL . (43c) 

0 

After this point, everything, including the resulting integral 

22 
'---~----- --- - --

equations and the solution method, is exactly the same as explained 

in Sec-Uon 2.2. After solving the aqua tions, the funct:ions G1 and G2 

are again substituted to (zr7) and (28) to find IS and m, which are 

then subst:ituted to (39) to find the two stress intensity factors at 

the corresponding value of x. 

2.4 ~etr.1c Internal Crack 

This is a special case of an internal crack where d=O. Due to 

the fact that simple tension cannot create any crack surface 

rotation and simple bending cannot create any crack opening 

displacement (on the neutral plane), from (23) and (24) one can 

write 

( 44) 

which gives 

~ tb = ~bt = 0 (45) 

Substitution of (45) into (37) would decouple the integral aqua. t:ions 

23 
(only unknown in (37a) is G1 while the only unknown in (37b) is G2) 

which would in turn enable us to solve them seperately. This 

property reduces both the computer time and memory space needed in 

the numerical solution. 

2.5 Two Opposite Elliptic Edge Cracks 

Two opposite edge cracks shown in Fig. 4 has exactly the same 

properties as symmetric internal cracks except for the coefficients 

of the shape functions. Once the shape functions are established by 

a suitable curve fitting, the same procedure that is applied to 

symmetric internal cracks is applied to find the stress intensity 

factors. 

2.6 Interaction Between Two Identical Cracks 

All the crack geometries that we considered up to now were 

single cracks lying in the region (-a, a) on the x1 axis. Only 

difference among them was their configuration in the thickness 

24 
direotion. Shape funotions, energy available for fraoture and 

funotions Uij and .., 1J are all sensitive to this configuration, 

while the form of the integral aqua. tion is independent of it. In 

this seotion, we will modify (1) and (2) for two identioal cracks 

which is shown in Fig. 5. In this case D is defined as (b/a , cia). 

With the new definition of D, (1) and (2) will take the following 

form. 

c' 

er(x') - ~f[~+ E 21r z-x' 

b' 

1 ero - ]G1(z) dz =-, 

z+x' E 

b' < x' < 0' , (46) 

m(x' ) 

E 

b' < x' < 0' ( 41) 

6E 

where 

b' = b/a, c r = cIa • 

Again we have to oonvert the limits of integrals to (-1,1) for 

25 
numerical solution purposes. By introducing the following parameters 

(keep in mind that from Fig. 5, (c'-b')/2 = 1 ) 

x = (2x'-c'-b') / (c'-b') (48) 

x' = [(c'-b')x+c'+b'] / 2 = x + [(c'+b') / 2] (49) 

dx '= dx 

and 

t = (2z-c ' -b ') / (c' -b' ) ( SO) 

z = [(c'-b')t+c'+b'] / 2 = t + [(c'+b') / 2] (51) 

dz = dt 

and substituting (27) and (28) for ~(x) and m(x) respectively, (46) 

and (47) can be rewritten as follows: 

x 

.... bb(x) J G,(t) 

-I 

1 

+ R4[-- K2(p1Y1 I) Y, 

26 

x 

dt + ~bt(x) J G2 (t) 

-I 

dt 
=- -1 < x < 1 ( 52) 

6E 

er o =- -1 < x < 1 (53) 

E 

where 

12 = t + x + b t + c' ( 54) 

Integral equations (52) and (53) will again be solved by GaussClebyshev closed form integration formula with the additional 

conditions 

1 f G1 (t) 

-I 

dt = 0 

1 f ~(t) dt = 0 

-I 

and the same backward procedure defined in Section 2.2 is used to 

find stress intensity factors at any point on the crack defined in 

27 
the region (b, c). Due to symmetry with respect to the %2%3 plane, 

stress intensity factors on the two cracks are also symmetric with 

respect to the same plane. Integral equations are valid for any two 

identical crack (edge, internal, two opposite) problem providing the 

corresponding ~ functions defined in Section 2.2 • 

2.7 Interaction of Three Identical Cracks 

As stated earlier integral equations for general case given in 

( , ) and (2) are valid for any number of cracks provided there is 

symmetry with respect to the %2X3 plane. This means that any three 

crack system composed of a symmetric crack surrounded. by two 

identical cracks (not needed to be identical with the third one) can 

be solved by using (1) and (2). But in this study, for 

simplification purposes, all of them are chosen identical. 

Configuration and parameters are shown in Fig. 6. D is defined as 

(0,1); (b/a , cIa). Because of the nature of the problem, we expect 

that crack surface displacements and rotations be symmetric with 

respect to the %2X3 plane. This suggests that two outer cracks 

should have the same rotation and displacement at the eqUidistant 

points from the origin, while the third one undergoes completely 

different displacement and rotation. For this reason G functions 

28 
------- -- -----------

should be defined seperately for inner crack and two outer cracks. 

Let 

Crack surface rotation derivative for 

the inner crack, (-a,a), 

Crack opening displacement derivative for 

the inner crack, (-a,a), 

Crack surface rotation derivative for 

the outer crack, (b,c), 

Crack opening displacement derivative for 

the outer crack, (b,c). 

Also <r (x) and m(x) for the inner and outer cracks should be defined 

seperately as follows, 

<r 1(x) : <rex) for inner crack, 

m1 (x) m(x) for inner crack, 

<r2 (x) <rex) for outer crack, 

lD:2(x) : m(x) for outer crack. 

(1) and (2) can be written for inner crack 

I 

_<r_1(_x_') - ~ f[ ~+ ~] G

11 (z) 

E 21r z-x' z+x' 

o 

29 
1 +-] 

z+x' 

er o =-

E 

o 

1 

+ R4[---- K2(p z-x' ) 

z-x' 

1 

o < x' < 1 , 

1 

+ R4[ - ~(pl z-x' I) 

z-x' 

+-

z+x' 

=- o < x' < 1 , 

6E 

and for outer crack 

I c· 

er 2(x') 1 J 1 1 

- - [-+ - ]G11 (Z) 

E 21r z-x' z+x ' 

dz --J[~ 21r z-x' 

o 

1 

+ - ] G12( z) dz = 

z+x, 

b 

b' < x' < c' , 

30 

( 55) 

( 56) 

( 57) 
~(x') 

6E 

1 1 

+ R4[- '2 ( p I z-x ' I ) +-

z-x' z+x' 

c· 

R1 f ~ ~-+- 1 1 

[z-x, z+x'] 

- R3[ 

b' 

1 1 

+ R4[- '2 (pi z-x I I> +-

Z_X' Z+X' 

'2(plz+x ' J)]] G12(z) 

1 1 

(z-x 1)3 + (Z+X I )3] 

~(plz+xII)]] ~2(z) 

=- b ' < X' < 0' , 

6E 

where X'= x/a, b'= bfa, c'=c/a. 

From (36), we may write 

1 

J [z~x. ={ + z:) 

1 

G11 (z) dz G11 (z) dz 

z-x' 

0 -I 

I 

1 1 1 1 

dz 

dz 

( 58) 

( 59) 

f [&z[z~x. +- - R3[ + ] + R4[- K2 ( p 1 z-x' I) 

z+x' (z-x' )3 (z+x,)3 z-x' 

0 

31 
-1 

+-

z+x' 

-I 

+ R4 ~ !2(P!Z-X'I)] G12(z) dz 

z-x' 

(60) 

Same expressions as (59) and (60) can also be written for G21 and 

~2· 

If we define 

x' = x for -1 < x' < 1 

z = t for -1 < z < 1 

and use (49) and (51) for b' < x' < c' and b' < z < c' 

respectively, and substitute (27), (28) for ~(x) and m(x), (55), 

(56), (57), (58) will take the following final form. 

x 

~bb(x) f G21 (t) dt -

-I 

32 
)C 

!bb{x) J G11 {t) dt -

-I 

6E 

:-

6E 

" 

G,,(t) dt + 'l'bt(x) J G:!2(t) dt 

-/ 

-1 < x < 1 , (61) 

, -1 < x < 1, (62) 

-1 < x < 1 , (63) 

33 
where Y1 

Y2 

Y3 

Y4 

YS 

dt - - - G 1 f~ 12(t) 

2. Y1 

0'0 

dt =- , E 

-I 

= t-x 

= t+x+b'+c' 

= t-x - [(b '+c') / 2] 

= t+x + [(b '+c') / 2] 

= t-x + [(b'+c') / 2] 

-1 < x < 1 , (64) 

This problem has four integral equa tions, (61), (62), (63), 

(64), with four unknown functions, G11 , G12, G21 , G22 , which will be 

sol.lJ'ed by Gauss-Chebyshev closed type integra tion formula under the 

follOWing single-valuedness conditions. 

1 f G11 (t) dt = 0 

-I 1 f G12 (t) dt = 0 

-I 

1 f G21 (t) dt = 0 

-I 

I J ~2(t) dt = 0 

-I 

After solving for the unknown functions, the same baclGlard procedur e 

34 
defined in Section 2.2 will be used for each crack seperately in 

order to find stress intensi ty factor's. 

Provided the corresponding It( functions, equations (61), (62), 

(63), (64) can be applied to any type (edge, internal, collinear) of 

identical cracks. The only reason for considering identical cracks 

to demonstrate interaction of three cracks in this section is 

simplicity. Line-spring model can also be applied to any three crack 

system which has two outer identical cracks with another 

symmetric(with respect to the X:2x3 plane) crack in the middle. In 

this case, It( functions and crack length will be def !ned seperately 

, 

for both(outer ones and inner one) cracks. One shouid keep in mind 

that crack length is a parameter in the coefficients of (61), (62), 

(63.), (64). Attention must be paid to redifine the coefficients in 

such case. 

35 
3. RESULTS AND DISCUSSION 

There are two main objectives in this study: 

1. Assess the accuracy of line spring model, 

2. Provide extensive numerical results which may be useful 

in application. 

Due to unavailability of the solutions for the cases that we 

are interested in, the first objective can only be reached for the 

central (d/h=O) elliptic internal crack under pure tension. 

Extensive results for central, eccentric internal cracks and two 

opposite edge cracks as well as the interaction among two and three 

cracks are tabulated. In all cases it was assumed that P=0.3 • 

However, the effect of P on the stress intensity factors does not 

seem to be significant. 

First step for the application of line spring model is to 

represent the SIFs as a polynomial in L/h. For this purpose, 

extensive information about plane strain (a/Lo ) stress intensity 

factors are needed. These results are obtained from previous 

materials for internal cracks, from [18] for two opposite edge 

cracks and are tabulated in Tables 1,3,5,7,9,11,13,15. Corresponding 

coefficients of shape functions are tabulated in Tables 

2,4,6,8,10,12,14,16. Coefficients of shape functions for the edge 

crack case are directly taken from [1] and tabulated in Table 17. 

36 
As noted before, for the application of the line spring model, 

the contour of the part-through crack can be any reasonable curve. 

Elliptic cracks are studied here since it is believed that ellipse 

is the closest contour for the actual shape of the crack which may 

be encountered in practical applications. Thus, crack length for any 

cross section is defined by, 

Lo being the total crack length at the midsection (x=O) • Note that 

the limiting values of the SIF are 

K-O for a/h-O 

The first case that is studied is the central internal elliptic 

crack ( d/h=O ) • Extensive results and formulas developed from a 

finite element method for this case are given in [19]. Table 18 and 

Figs. 19,20 show the comparision of the SIFs obtained from this 

study wi th those genera ted from the formulas given in [19]. SIFs 

obtained from [19] are represented as Kr while the normalizing SIF 

GO where is 0 = N22/h. Both values and the percent 

differences between them are given for various parameters at x=O and 

x=1/2 • As expected, the SIFs K(Lo) for the elliptic crack are 

37 
consistently smaller than the plane stral.n values. The table shows 

that with the exception of relatively small values of alLo at small 

Lo/h (for which the line spring is really not a suitable model) the 

agreement is surprisingly good. From the results, one may conclude 

that line spring can be used with some confidence for any central 

internal elliptic crack which has an a/Lo ratio larger than 2. 

SIFs for the same geometry, but under pure bending are given in 

Table 20 (Fig. 21). This time, normalizing SIF is defined as 

Ko = mo(1rLo/2) 1/2 where mo = 6~~/h • The results given are for the 

tension side. On the compression side the stress intensity factors 

have the same values with a negative sig~ Under pure bending, since 

the crack faces on the compression side of the plate would close, 

th~ results given in the table cannot be used seperately. The 

resul ts are, of course, useful and valid if the plate is subjected 

to tenSion, as well as bending, in such a way that the superimposed 

SIF is positive everywhere. 

From Table 20, by excluding the case of a/Lo=0.5 which proved 

to be unreliable, it may be concluded that SIF are independent of 

a/Lo ratio for Lo/h < 0.5 • For Lo/h > 0.5 , SIF begins to increase 

with increasing Lo/h ratio, which is expected. Though we are unable 

to confirm the accuracy of the line spring model in this case, we 

could say that there is no unusual behaviour in the results. Results 

are asymptotically approaching to plane strain values as a/Lo is 

38 
increased except for large Lo/h values. 

Another comparision with the previous finite element results 

[20] is shown in Table 19 (Fig. 23). It should be noted that in the 

results given in Table 19 a/Lo=1.25 is relatively small for the line 

spring model to be effective. Despite that, the relative error does 

not seem to be very high. 

After symmetric internal cracks, nons ymme tri cally located 

internal cracks are studied both under pure tension and pure bending 

conditions. Extensive results which scan almost all possible 

configurations ar~ given in Tables 21-32 (Figs. 7-18). Normalizing 

SIFs are the same as that defined in the previous case for both 

tension and bending. Again we are unable to verify in which region 

the line spring results are reliable because of the lack of previous 

work done on this kind of problem. AsymptotiC convergence to plane 

strain results is fairly good. A close observation of the tension 

resul ts would show that for small a/Lo ratios SIFs first decrease, 

then increase with increasl.ng Lo/h. This is mainly due to the fact 

that tabulated results are normalized with respect to ~ oV1fLo/2 

Thus, even though the normalized SIF is decreaSing, the real SIF may 

increase (which indeed happens in this case) but with a lower rate 

than the rate of increase of Lo • Also, again for small Lo/h and 

a/Lo values SIF on the inside part of the crack front turns out to 

be greater than that on the other part of the crack border which is 

39 
closer to the free surface. 

Two opposite elliptic edge cracks have been studied only under 

pure tension. Normalizing stress intensity factor is again 

er o (1I'Lo/2)1/2 with er o = N~/h. Decreasing-increasing behaviour of 

the SIFs is again observed here. Same explanation which is given 

earlier is also valid for this case. Plane strain results are very 

close to each other in the region that we observe the slight 

decrease in SIFs. For this reason, a small uncertaini ty in the 

calculations can easily result with a slight decrease while we are 

expecting a slight increase. Except this behaviour, everything is as 

expected in Table 33 (Fig. 22). 

Resul ts for the interaction of two identical elhptic internal 

cracks under pure bending and pure tension have been tabulated in 

Tables 34 and 35 respectively. It can be seen that there is almost 

no interaction between two identical elliptic edge cracks if the 

distance between them (Fig.S) is larger than 8 times the half crack 

length, a. Another result which may be observed is that the 

interaction is more effective for the cracks having large Lo/h 

ratios. 

SIFs for the two identical central internal elliptic cracks are 

tabulated in Table 36. Results are given only for tension case 

because it has been observed that there is no appreciable 

40 
---------- ---

interaction for the crack geanetries given in Table 36 under pure 

bending. SIraeviates not more than 1% fran the single crack value 

for b/a=0.1 in bending. It is seen that as the distance between the 

two cracks is increased the single crack solution is easily 

recovered. There is almost m interaction for bla ~ 0.1 for the 

cracks Lolh < 0.5 • If Lolh > 0.5 , no interaction region can be 

defined for bla ~ 4 • 

Interaction results for any two identical elliptic internal 

cracks are given in Table 37. Sane erra tic results are observed for 

small alLo ratios at small Lolh ratios. Clearly, for these crack 

i geanetries line spring is not a suitable model. 

_ As the ~ast example, SIFs for the three identical internal 

elliptic cracks under pure tension are cal c ula ted and are given in 

Table 38. The only conclusion one may draw from these results is 

that the SIFs on the middle crack are slightly higher than the SIFs 

on the outer crack. But the difference is so small that they may be 

regarded as eq ual. 

41 
TAB L E S 

42 
" 

Table 1. Stress intensity factors for centrally cracked 

plate subjected to tension N or bending M 

under plane strain conditions. 

Llh (K/Ko)N (K/Ko)M 

0.05 0.0250 

0.1 1.0060 0.0500 

0.2 1.0246 0.1001 

0.3 1.0577 0.1505 

0.4 1.1094 0.2023 

0.5 1.1867 0.2573 

0.6 1.3033 0.3197 

0.7 1.4884 0.3986 

0.8 1.8169 0.5186 

0.9 2.585 0.7776 

0.95 4.252 1.1421 

_ Table 2. The coefficients bi and ci for the shape 

. functions gt and gb for symmetric internal 

(d/h=O) crack. 

i bi ci 

1 0.7070 0.0169 

2 0.4325 -0.4629 

3 -0.1091 15.0622 

4 7.3711 -143.7384 

5 -57.7894 807.2449 

6 271.1551 -2844.8525 

7 -744.4204 6468.9152 

8 1183.9529 -9477.5512 

9 -1001.4920 a538.7826 

10 347.9786 -4455.2167 

11 993.2482 

43 
Table 3. Stress intensi ty factors for the plate which has an 

internal crack with d/h:0.05 subjected to uniform 

tension N or bending M under plane strain aondi tions. 

Llh (KA/Ko)N (KB/Ko)N (KA/Ko) M (KB/Ko)M 

0.0001 0.1001 0.1000 

0.09 1.0053 1.0050 0.1455 0.0555 

0.18 1.0221 1.0200 0.1923 0.0120 

0.27 1.0530 1.0452 0.2409 -0.0305 

0.36 1.1021 1.0823 0.2925 -0.0722 

0.45 1.1769 1.1340 0.34 (]l -0.1134 

0.54 1.2909 1.2053 0.4175 -0.1547 

0.63 1.4731 1.3053 0.5068 -0.1 (]l3 

0.72 1.7983 1.4518 0.6467 -0.2430 

0.81 2.5631 1.6887 0.9525 -0.2923 

0.855 3.6610 1.8858 1.3814 -0.3125 

Table 4. The coefficients of bi and ci for the shape 

functions gt and gb for internal crack with 

d/h:0.05. 

i bAi ~i cAi cBi 

1 0.7071 0.7071 0.0708 0.0707 

2 0.45 (]l 0.4347 -0.0623 -0.3701 

3 0.7671 -0.0915 13.1229 0.5654 

4 0.1552 2.6(]l3 -166.4280 -6.6423 

5 -9.3017 -14.1195 1145.8217 45.7189 

6- (]l.3172 54.9653 -4762.0914 -189.9515 

7 -413.9673 -135.3432 12511.5152 498.8463 

8 936.4719 205.3051 -20927 .001 9 -834.5704 

9 -1078.2322 -173.3480 21613.9362 862.1672 

10 504.0555 62.8847 -12568.0268 -501.4354 

11 3148.4879 125.586 9 

44 
Table 5. Stress intensi ty factors for the plate which has an 

internal crack with d/h:0.10 subjected to uniform 

tension N or bending M under plane strain condi tiona. 

Llh (KA/Ko)N (KB/Ko)N (KA/Ko)M (KB/Ko)M 

0.0001 0.2001 0.2000 

0.08 1.0050 1.0046 0.2410 0.1609 

0.16 1.0212 1.0179 0.2843 0.1236 

0.24 1.0513 1.0399 0.3308 0.0881 

0.32 1.0998 1.0709 0.3821 0.0545 

0.40 1.1743 1.1126 0.4410 0.0231 

0.48 1.2887 1.1677 0.5136 -0.0059 

0.56 1.4722 1.2418 0.6123 -0.0313 

0.64 1.8002 1.3465 0.7701 -0.0504 

0.72 2.5705 1.5167 1.1183 -0.0519 

0.76 3.6693 1.6727 1.6050 -0.0280 

Table 6. The coefficients of bi and ci for the shape 

functions gt and gb for internal crack with 

d/h:0.10. 

i bAi bai cAi cBi 

1 0.7071 0.7072 0.1415 0.1414 

2 '0.5498 0.5043 -0.1734 -0.3871 

3 1.5235 -0.5779 18.7434 1.2936 

4 -2.2395 7.6480 -266.7713 -17.0715 

5 -5.2844 -52.8793 2066.4692 132.0282 

6- 226.0267 257 .2074 -9661.5218 -617.3023 

7 -1423.2887 -799.7410 28556.2764 1826.3191 

8 4348.1446 1530.8314 -53734.1216 -3441.9797 

9 -6553.5540 -1634.0240 62435.9340 4007.6642 

10 3959.2116 749.0673 -40844.2364 -2628.6642 

11 11511.5912 743.3343 

45 
Table 7. Stress intensity factors for the plate which has an 

internal crack with dlh=0.15 subjected to uniform 

tension N or bending M under plane strain conditions. 

Llh (IC A/ICo)N (ICB/Ko)M (ICA/ICo)M (KB/ICo)M 

0.0001 0.3001 0.3000 

0.07 1.0049 1.0045 0.3365 0.2663 

0.14 1.0208 1.0172 0.3763 0.2352 

0.21 1.0506 1.0380 0.4206 0.2066 

0.28 1.0988 1.0671 0.4714 0.1807 

0.35 1.1729 1.1057 0.5319 0.1500 

0.42 1.2868 1.1561 0.6089 0.1395 

0.49 1.4693 1.2235 0.7164 0.1271 

0.56 1 .. 7938 1.3197 0.89>9 0.1259 

0.63 2.5476 1.4~9 1.2760 0.1533 

0.665 3.6065 1.6426 1.0090 0.2032 

Table 8. The coefficients of bi and 0i for the shape 

functions gt and gb for internal craok with 

d/h=0.15. 

i bAi bBi cAi oBi 

1 0.7071 0.7072 0.2122 0.2121 

2 0.7028 0.6376 -0.2929 -0.4042 

3 2.7653 -1.2331 26.3239 2.2494 

4 -7.2036 19.0057 -427 .255 8 -33.6757 

5 9.1384 -173.8407 3782.9591 2gr.499> 

6 667.4954 1108.9410 -20214.1250 -159>.1109 

7 -6105.7233 -4517.1019 68285.5344 5378.6049 

8 25260.2847 11317.3469 -146859.5866 -11588.2217 

9 -50586.0954 -15002.5485 195038.2341 15425.1699 

10 40325.8388 9475.7400 -145833.6228 -11566.8288 

11 46900.5243 3740.3538 

46 
Table 9. Stress intensity factors for the plate which has an 

internal crack with d/h=0.20 subjected to uniform 

tension N or bending M under plane strain conditions. 

Llh (KA/Ko)N (KB/Ko)N (KA/Ko)M (KB/Ko)M 

0.0001 0.4001 0.4000 

0.06 1.0048 1.0044 0.4319 0.3718 

0.12 1.0205 1.0170 0.4682 0.3468 

0.18 1.0497 1.0374 0.5102 0.3252 

0.24 1.0968 1.0660 0.5601 0.3070 

0.30 1.1691 1.1034 0.6216 0.2930 

0.36 1.2799 1.1535 0.7020 0.2844 

0.42 1.4562 1.2202 0.8162 0.2839 

0.48 1.7668 1.3161 1.0031 0.2981 

0.54 2.4756 1.4006 1.4127 0.3482 

0.57 3.4498 1.6422 1. 9674 0.4165 

Table 10. The coefficients of bi and ci for the shape 

functions gt and gb for internal crack with 

d/h=0.20. 

i bAi bsi cAi cBi 

1 0.7071 0.7072 0.2829 0.2828 

2 0.9394 0.8534 -0.4105 -0.4192 

3 5.0186 -2.2518 36.3675 3.4524 

4 -19.6345 47.2610 -686.9924 -59.384 8 

5 76.1489 -589.3736 7097 .1745 611.7194 

6 2376.8770 5125.5432 -44245.1037 -3814.7743 

7 -32402.0663 -28413.7181 174386.6733 15058.0019 

8 187563.3073 96 818. 0664 -437594.1661 -37850.22 90 

9 -517758.7465 -183743.9141 678087.6506 58816.9514 

10 566112.6482 149736.5141 -591607.7634 -51479.6217 

11 222394.8277 19433.7456 

47 
Table 11. Stress intensity factors for the plate which has an 

internal crack with d/h=0.25 subjected to uniform 

tension N or bending M under plane strain conditio~ 

Llh (KA/Ko)N (KB/Ko)N (KA/Ko) M (KB/Ko)M 

0.0001 0.5001 0.5000 

0.05 1.0046 1.0042 0.5273 0.4771 

0.10 1.0197 1.0165 0.5598 0.4583 

0.15 1.0476 1.0364 0.5991 0.4434 

0.20 1.0925 1.0644 0.6473 0.4327 

0.25 1.1610 1.1016 0.70~ 0.4270 

0.30 1.2652 1.1505 0.7900 0.4277 

0.35 1.4295 1.2160 0.9074 0.4380 

0.40 1.7151 1.3099 1.0996 0.4649 

0.45 2.3529 1.46~ 1.5149 0.5310 

0.475 3.2077 1.6192 2.0658 0.6085 

Table 12. The coefficients of bi and c i for the shape 

functions gt and gb for internal crack with 

d/h=0.25. 

i bA1 bB1 cA1 cB1 

1 0.7071 0.7071 0.3536 0.3536 

2 1.3041 1.1918 -0.5246 -0.4284 

3 9.6510 -4.1793 50.2589 4.8592 

4 -57.8163 129.7358 -1136.1027 - 97.6558 

5 450.7761 -2325.3551 14085.0245 1206.5210 

6- 10351. 8621 29069.6053 -105370.0173 -~29. 9162 

7 -227 gr 3. 91 93 -231101.0011 4 98386 • 1 47 5 42788.3389 

8 1960136.6080 1120028.3541 -1500844.1305 -129153.3052 

9 -7899243.5583 -3063223.0225 2791092.0497 240895.9775 

10 12538854.7550 3568834.5097 -2922538.2779 -253169.5873 

11 13 1 85 91 • 9416 114779.2965 

48 
1 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

Table 13. Stress intensi ty fact or s for the plate which has an 

-

internal crack wi th d/h:0.30 subjected to uniform 

tension N or bending M under plane strain condi tiona. 

Vh (KA/Ko)N (KB/Ko)N (KA/Ko)M (KB/Ko)M 

0.001 0.6001 0.6000 

0.04 1.0043 1.0040 0.6226 0.5824 

0.08 1.0183 1.0155 0.6510 0.5686 

0.12 1.0442 1.0343 0.6867 0.56 rrr 

0.16 1.0855 1.0608 0.7320 0.5568 

0.20 1.1481 1.0959 0.790 9 0.5584 

0.24 1.2425 1.1419 0.8708 0.5669 

0.28 1.3898 1.2031 0.9866 0.5855 

0.32 1.6423 1.2895 1.1755 0.6209 

0.36 2.1937 1.4309 1.5773 0.6939 

0.38 2.9157 1.5591 2.0992 0.7709 

Table 14. The coefficients of bi and c1 for the shape 

functions gt and gb for internal crack with 

d/h:0.30. 

bAi bs1 cA1 cB1 

0.7071 0.7071 0.4244 0.4243 

1.S()27 1.7480 -0.6553 -0.4321 

20.8636 -8.9087 72.3179 6.6403 

-197.8895 440.7103 -2037.1553 -160.0429 

2675.1513 -12390.9668 31569.7505 2468.7747 

71601.3880 241718.2677 -295202.5115 -23090.8486 

-2639273.5314 -2989605.6659 1745346.790 8 136788.4159 

35994016.1511 22618755.1306 -6570139.7052 -516213.4074 

-227958779.2155 -94950523.9757 15273818.2302 120392 8.2026 

567162515.8604 170542044.2578 -19993047.6604 -1582260.0614 

11276922.9053 897240.7983 

49 
Table 15.-Stress intensity factors for the plate with 

collinear edge cracks subJected to uniform 

tension N under plane strain conditions. 

Uh (K/Ko) N 

0.0001 1.1221 

0.1 1.1231 

0.2 1.1254 

0.3 1.1292 

0.4 1. 1370 

0.5 1.1546 

0.6 1.2117 

0.7 1.3254 

0.8 1. 5393 

0.9 2.0836 

Table 16. The coefficients of bi for ~e shape function 

gt fO: collinear edge crack. 

i bi 

1 0.7934 

2 0.0775 

3 -0.7542 

4 7.5~5 

5 -12.1712 

6 -186.5011 

7 1236.2858 

8 -3043.6190 

9 3350.3456 

10 -1374.8426 

50 
Table 17. The coefficients of b1 and c1 for the shape 

functions gt and gb for edge crack. 

1 b1 c1 

1 1.1216 1.1202 

2 6.5200 -1.8872 

3 -12.3877 18.0143 

4 89.0554 -87 .3851 

5 -188.6080 241.9124 

6 207.3870 -319.9402 

7 -32.0524 168.0105 

51 
-~ - ----------~--

Table 18. Comparison of the stress intensity factors K{x) 

calculated in this study at x=O and x=1/2 (x=x 1/a) 

for an internal planar elliptic crack in a pla te under 

unifonn tension N with the corresponding values Kr{x) 

given in Hef.[19] • %D=100{Kr -K)/Kr • 

Lolh alLo x K{x)/Ko Kr{x)/Ko %D 

0.1 0.5 0 0.916 0.637 43.7 

0.1 0.5 1/2 0.868 0.637 36.2 

0.1 1.0 0 0.955 0.827 15.5 

0.1 1.0 112 0.896 0.785 14.2 

0.1 2.0 0 o. f]l6 0.935 4.3 

0.1 2.0 112 0.911 0.875 4.1 

0.1 3.0 0 0.983 0.967 1.7 

0.1 3.0 112 0.916 0.902 1.6 

0.1 4.0 0 0.987 0.980 0.6 

0.1 4.0 112 0.919 0.914 0.5 

0.1 10.0 0 0.993 0.999 -0.6 

0.1 10.0 112 0.923 0.930 -0.7 

0.2 0.5 0 o. E62 0.638 35.1 

0.2 0.5 1/2 0.827 0.638 29.6 

0.2 1.0 0 0.931 0.830 12.2 

0.2 1.0 112 0.880 0.788 11.6 

0.2 2.0 0 o. f]l1 0.942 3. 1 

, 0.2 2.0 112 0.908 0.881 3.1 

0.2 3.0 0 0.986 o. CJl'6 1.0 

0.2 3.0 1/2 0.918 0.910 0.9 

0.2 4.0 0 0.993 0.991 0.2 

0.2 4.0 1/2 0.923 0.923 0.0 

0.2 10.0 0 1.007 1.013 -0.6 

0.2 10.0 1/2 0.933 0.942 -1.0 

52 
Table 18- C~nt. 

Lo/h a/Lo x K(x)/Ko Kr(x)/Ko %D 

0.3 0.5 0 0.82!i 0.641 28.6 

0.3 0.5 112 0.796 0.640 2!i.4 

0.3 1.0 0 0.920 0.837 9.9 

0.3 1.0 112 0.871 0.79!i 9.8 

0.3 2.0 0 0.979 0.957 2.3 

0.3 2.0 112 0.914 0.893 2.3 

0.3 3.0 0 1.001 0.996 0.5 

0.3 3.0 112 0.930 0.927 0.3 

0.3 4.0 0 1.012 1.01!i. -0.2 

0.3 !i.0 112 0.937 0.942 -0.5 

0.3 10.0 0 1.034 1.041 -0.7 

0.3 10.0 112 0.952 0.966 -1.5 

0.4 0.5 0 0.798 0.645 23.6 

0.4 0.5 112 0.775 0.644 20.3 

0.4 1.0 0 0.920 0.851 8. 1 

0.4 1.0 1/2 0.871 0.804 8.3 

0.4 2.0 0 1.000 0.98!i 1.6 

0.4 2.0 112 0.929 0.915 1.6 

, 0.4 3.0 0 1.030 1.031 -0.1 

O.!i 3.0 112 0.950 0.955 -0.4 

0.4 4.0 0 1.047 1.054 -0.7 

0.4 4.0 1/2 0.961 0.974 -1.3 

0.4 10.0 0 1.078 1.091 -1.2 

0.4 10.0 1/2 0.982 1.005 -2.3 

53 
Table 18- Cont. 

Lo/h a/Lo x K(x)/Ko Kr(x)/Ko %D 

0.5 0.5 0 0.783 0.654 19.8 

0.5 0.5 1/2 0.761 0.650 17.2 

0.5 1.0 0 0.932 0.874 6.7 

0.5 1.0 1/2 0.880 0.821 7.3 

0.5 2.0 0 1.036 1.030 0.6 

0.5 2.0 1/2 0.956 0.949 0.7 

0.5 3.0 0 1.078 1.090 -1.1 

0.5 3.0 112 0.98!J 0.998 -1.4 

0.5 4.0 0 1.101 1.121 -1.8 

0.5 4.0 112 0.998 1.023 -2.4 

0.5 10.0 0 1.145 1.172 -2.4 

0.5 10.0 1/2 1.025 1.063 -3.5 

0.6 0.5 0 0.779 0.667 16.9 

0.6 0.5 112 0.756 0.658 14.9 

0.6 1.0 0 0.960 0.911 5.4 

0.6 1.0 1/2 0.901 0.846 6.4 

0.6 2.0 0 1.095 1.103 -0.8 

0.6 2.0 1/2 0.997 0.999 -0.2 

0.6 3.0 0 1.151 1.183 -2.7 

0.6 3.0 1/2 1.033 1.061 -2.6 

0.6 4.0 0 1.183 1.225 -3.5 

0.6 4.0 1/2 1.052 1.092 -3.6 

0.6 10.0 0 1.245 1.298 -4.0 

0.6 10.0 1/2 1.088 1.143 -4.7 

54 
Table 18- Cont. 

Lo/h alLo x K(x)/Ko Kr(x)/Ko %D 

0.7 0.5 0 0.788 0.687 14.7 

0.7 0.5 1/2 0.760 0.671 13.3 

0.7 1.0 0 1.009 0.968 4.3 

0.7 1.0 1/2 0.935 0.882 6.0 

0.7 2.0 0 1.187 1.213 -2.2 

0.7 2.0 1/2 1.058 1.067 -0.8 

0.7 3.0 0 1.266 1.322 -4.2 

0.7 3.0 1/2 1.106 1.144 -3.3 

0.7 4.0 0 1.310 1.381 -5.1 

0.7 li.O 1/2 1.132 1.183 -4.4 

0.7 10.0 0 1.403 1.483 -5.4 

0.7 10.0 1/2 1.179 1.242 -5.1 

0.8 0.5 0 0.818 0.717 14. 1 

0.8 0.5 1/2 0.777 0.686 12.9 

0.8 1.0 0 1.096 1.051 4.3 

0.8 1.0 1/2 0.991 0.930 6.6 

0.8 2.0 0 1. 341 1.372 -2.3 

0.8 2.0 1/2 1.152 1.152 0.0 

- 0.8 3.0 0 1.457 1.521 -4.3 

0.8 3.0 1/2 1.218 1.245 -2.2 

0.8 4.0 0 1.525 1.603 -4.9 

0.8 4.0 1/2 1.253 1.290 -2.9 

0.8 10.0 0 1.674 1.747 -4.2 

0.8 10.0 1/2 1.318 1.349 -2.3 

55 
Table 18- Cont. 

Lo/h a/Lo x K(x)/Ko Kr(x)/Ko %D 

0.9 0.5 0 O. ~5 0.759 19.3 

0.9 0.5 112 0.813 0.709 14.7 

0.9 1.0 0 1.284 1.167 10.0 

0.9 1.0 1/2 1.086 0.987 10.1 

0.9 2.0 0 1.661 1.594 4.2 

0.9 2.0 112 1.310 1.246 5.1 

0.9 3.0 0 1.858 1.799 3.3 

0.9 3.0 112 1.405 1.347 4.2 

0.9 4.0 0 1.981 1.912 3.6 

0.9 4.0 112 1.456 1.392 4.6 

Table 19. Comparison of the stress intensity factors K(Lo ) 

calculated in this study at the midsection of a 

symmetrically loca ted internal(d/h=O) planar crack 

in a plate under uniform tension N with the corres~onding results Kr given in [20J. Lo/h=.75, 

a/Lo= 1.25, x=xl la=cosi J %D=1 OO(K_Kr)/Kr • 

8 x Kr/K 

0 K/Ko %D 

90 0 0 0.985 1.120 13.7 

800 0.17!j 0.971 1.103 13.6 

700 0.342 0.944 1.052 11.4 

600 0.500 0.898 0.973 8.4 

450 0.707 0.810 0.832 2.7 

!jOo 0.766 0.770 0.742 -3.6 

56 
Table 20. Normalized stress intensi ty factors calculated 

at the midsection of a symmetrically located 

(d/b=O) internal planar elliptic crack in a 

plate under uniform bending Me 

a/Lo-,0.5 2.0 3.0 4.0 5.0 10.0 Plane 

LO/b Strain 

0.1 .050 .050 .050 .050 .050 .050 .050 

0.2 .099 .099 .099 .099 .099 .099 .100 

0.3 .148 .149 .149 .149 .149 .149 .151 

0.4 .195 .199 .199 .199 .200 .200 .202 

0.5 .240 .250 .251 .252 .253 .254 .257 

0.6 .285 .305 .308 .310 .311 .313 .320 

0.7 .332 .369 .375 .379 .382 .387 .399 

0.8 .387 •• 457 .470 .478 .483 .496 .519 

0.9 .476 .619 .650 .670 .683 .717 .778 

57 
Table 21. Normalized stress intensi ty factors cal culated at the 

midsection of an internal planar elliptic crack with 

d/h=.05 in a plate under uniform tension N. 

KAIKo 

a/Lo_0.5 1.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lolh Strain 

.05 0.954 0.975 0.987 0.990 0.992 0.995 0.996 1.002 

.15 0.887 0.944 O. gr6 0.988 0.994 1.001 1.004 1.015 

.25 0.838 0.926 0.979 0.998 1.008 1.022 1.028 1.045 

.35 0.804 0.921 0.996 1.025 1.040 1.061 1.070 1.0!:6 

.45· 0.782 0.931 1.032 1.073 1.095 1.126 1.139 1.177 

.55 0.775 0.959 1.096 1.153 1.185 1.231 1.251 1.307 

.65 0.786 1.018 1.204 1.288 1.336 1.407 1.439 1.529 

.75 0.828 1.134 1.407 1.539 1.619 1.742 1. 799 1. (]TO 

.85 0.994 1.479 1.987 2.268 2.452 2.761 2.921 3.481 

a/Lo_0.5 1.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lolh Strain 

.05 0.955 0.976 0.987 0.991 0.993 0.995 0.996 1.002 

.15 0.891 0.946 o. (]T7 0.988 0.993 1.001 1.003 1.014 

.25 0.847 0.929 0.978 0.996 1.006 1.018 1.023 1.039 

.35 - 0.817 0.923 0.991 1.017 1.031 1.049 1.056 1.078 

.45 0.798 0.928 1.017 1.051 1.070 1.095 1.105 1.134 

.55 0.789 0.944 1.056 1.102 1.127 1. 161 1.176 1.215 

.65 0.788 0.972 1.115 1.176 1.210 1.257 1.277 1.333 

.75 0.797 1.017 1.201 1.284 1.332 1.401 1.432 1.517 

.85 0.822 1.088 1.334 1.455 1.529 1.643 1.697 1.859 

58 
Table 22. Normal--j:z-ed stress intensity factors calculated at the 

midsection of an internal planar elliptic crack wi tli 

d/h=0.10 in a plate under uniform tension N. 

KAIKo 

a/Lo_0.5 1.0 2.0 3.0 4.0 7.0 1Q.0 Plane 

Lolh Strain 

.05 0.950 0.974 0.986 0.990 0.992 0.995 0.996 1.002 

.15 0.878 0.940 O. gr5 0.988 0.994 1.003 1.006 1.018 

.25 0.825 0.921 0.980 1.002 1.013 1.029 1.036 1.056 

.35 0.789 0.919 1.004 1.037 1.055 1.081 1.092 1.124 

.45 0.770 0.937 1.056 1.105 1.132 1.172 1.189 1.240 

.55 0.770 0.986 1.152 1.226 1.268 1.331 1.360 1.444 

.65 0.805 1.093 1.342 1.462 1.534 1.646 1.699 1.860 

.75 O. ~1 1.426 1.900 2.159 2.329 2.615 2.764 3.291 

KBIKo 

a/Lo-0.5 1.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lo/h Strain 

.05 0.952 0.974 0.986 0.990 0.993 0.995 0.996 1.002 

.15 0.886 0.943 o. gr6 0.988 0.993 1.001 1.005 1.016 

.25 0.843 0.927 0.979 0.998 1.007 1.021 1.026 1.043 

.35 0.815 0.923 0.993 1.020 1.034 1.054 1.062 1.085 

.45 0.798 0.928 1.019 1.055 1.075 1.102 1.113 1.145 

.55 . 0.789 0.944 1.05"9 1.107 1.133 1. 171 1.187 1.231 

.65 0.788 0.970 1.116 1.181 1.218 1.272 1.296 1.363 

.75 0.795 1.010 1.203 1.297 1.355 1.445 1.488 1.625 

59 
Table 23. Normalized stress intensity factors calculated at the 

midsection of an internal planar elliptic crack with 

d/h::0.15 in a plate under uniform tension N. 

KA/Ko 

a/Lo-0.5 1.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lolh Strain 

.05 0.944 0.971 0.985 0.989 0.992 0.995 0.996 1.002 

.15 0.864 0.934 0.974 0.988 0.996 1.006 1.010 1.024 

.25 0.808 0.916 O. 9 Btl 1.009 1.022 1.041 1.049 1.076 

.35 0.773 0.921 1.020 1.060 1.083 1.115 1.129 1.173 

.45 0.761 0.956 1.101 1.164 1.200 1.254 1.279 1.354 

.55 0.784 1.051 1.273 1.378 1.441 1.540 1.588 1.733 

.65 0.926 1.362 1. 789 2.021 2.172 2.427 2.560 3.033 

KBlKo 

a/Lo-0.5 1.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lolh Strain 

.05 0.964 0.972 0.985 0.990 0.992 0.995 0.996 1.002 

.15 O. Err6 0.939 O. q[5 0.988 0.994 1.003 1.007 1.020 

.25 0.833 0.923 0.979 1.000 1.012 1.027 1.033 1.054 

.35 0.806 0.920 0.997 1.027 1.044 1.067 1.077 1.106 

.45 0.791 0.929 1.028 1.070 1.093 1.126 1.141 1.182 

.55 0.784 0.947 1.076 1.134 1.167 1.216 1.238 1.303 

.65 0.783 0.976 1.152 1.239 1.293 1.379 1.422 1.561 

60 
Table 24. Normalized stress intensity factors calculated at the 

midsection of an internal planar elliptic crack with 

d/h=0.20 in a plate under uniform tension N. 

KAIKo 

a/Lo-D.5 1.0 2.0 3.0 4.0 1.0 10.0 Plane 

Lolh Strain 

.05 0.964 0.967 0.983 0.989 0.991 0.995 0.996 1.003 

.15 0.848 0.928 0.914 0.990 0.999 1.011 1.016 1.033 

.25 0.790 0.914 0.992 1.022 1.039 1.062 1.073 1.107 

.35 0.761 0.933 1.054 1.104 1.133 1.176 1.196 1.258 

.45 0.110 1.019 1.202 1.289 1.342 1.424 1.463 1.5~ 

.55 0.896 1.293 1.666 1.863 1.990 2.204 2.316 2.110 

KB/Ko 

- a/Lo~O. 5 1.0 2.0 3.0 4.0 1.0 10.0 Plane 

Lolh Strain 

.05 0.939 0.968 0.984 0.989 0.992 0.995 0.996 1.003 

.15 0.863 0.933 0.914 0.988 0.996 1.006 1.011 1.026 

.25 0.818 0.918 0.982 1.006 1.020 1.038 1.046 1.072 

.35 0.793 0.918 1.007 1.043 1.063 1.092 1.105 1.144 

.45 0.179 0.931 1.049 1.102 1.132 1.11 B 1.199 1.263 

.55 0.710 0.955 1.121 1.204 1.256 1.340 1.382 1.524 

61 
Table 25. Normalized stress intensity factors calculated at the 

midsection of an internal planar elliptic crack with 

d/h::0.25 in a plate under uniform tension N. 

KAIKo 

a/Lo-0.5 1.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lo/h Strain 

.05 0.927 0.962 0.981 0.988 0.991 0.995 0.997 1.005 

.15 0.831 0.922 O. CJf6 0.995 1.006 1.020 1.026 1.048 

.25 0.776 0.919 1.013 1.051 1.071 1.101 1.115 1.161 

.35 0.768 O. CJf7 1.133 1.201 1.241 1.303 1.333 1.429 

.115 0.876 1.228 1.5110 1.698 1.798 1.9611 2.051 2.353 

KB/Ko 

a/Lo_0.5 1.0 2.0 3.0 4.0 7.0 10.0 Plane 

1" Ih Stral.n 0 

.05 0.930 0.964 0.982 0.988 0.991 0.995 0.997 1.004 

.15 0.847 0.926 O. CJf3 0.991 1.000 1.012 1.017 1.036 

.25 0.801 0.913 0.989 1.018 1.034 1.057 1.068 1.102 

.35 0.775 o. 919 1.026 1.072 1.098 1.138 1.157 1.216 

.45 0.758 0.938 1.095 1.172 1.220 1.2CJf 1.336 1.468 

62 
Table 26. Normalized stress intensity factors calculated at the 

midsection of an internal planar elliptic crack wi th 

d/h=0.30 in a plate under uniform tension N. 

KAlKa 

a/Lo ... 0.5 1.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lo/h Strain 

.05 0.917 0.958 0.980 0.987 0.991 0.996 0.998 1.007 

.15 0.816 0.921 0.985 1.008 1.021 1.038 1.046 1. 073 

.25 0.700 0.951 1.070 1.119 1.146 1.187 1.207 1.273 

.35 0.869 1.170 1.420 1.537 1.610 1.727 1.7f!7 1.995 

KB/Ko 

a/Lo-0.5 1.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lo/h Strain 

.05 0.919 0.959 0.900 0.987 0.991 0.996 0.997 1.006 

.15 0.829 0.920 O. ':Jl6 0.997 1.008 1.023 1.030 1.053 

.25 0.783 0.914 1.005 1.042 1.063 1.093 1.108 1.156 

.35 0.753 0.927 1.071 1.138 1.178 1.243 1.276 1.3f!7 

63 
Table 27. Normalized stress intensity factors calculated at the 

midsection of an internal planar elliptic crack wi th 

d/h=0.05 in a plate under uniform bending M. 

KAIKo 

a/Lo-l.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lolh Strain 

.05 0.118 0.119 0.120 0.120 0.120 0.120 0.121 

.15 0.170 0.173 0.174 0.175 0.176 0.176 O. 177 

.25 0.217 0.222 0.224 0.225 0.227 0.228 0.230 

.35 0.266 0.274 0.278 0.279 0.282 0.283 0.2Er7 

.45 0.316 0.329 0.335 0.338 0.342 0.344 0.350 

.55 0.371 0.391 0.400 0.405 0.412 0.416 0.426 

.65 0.436 0.469 0.484 0.493 0.506 0.513 0.532 

.75 0.528 0.5e6 0.614 0.631 0.659 0.673 0.716 

.85 0.753 0.889 0.963 1.013 1.099 1.144 1.313 

KB/Ko 

a/Lo-l.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lo/h Strain 

.05 0.073 0.074 0.074 0.074 0.074 0.075 0.075 

:15 0.020 0.023 0.024 0.025 0.025 0.026 0.026 

.25 -0.032 -0.027 -0.025 -0.024 -0.023 -0.022 -0.021 

.35 -0.082 -0.075 -0.073 -0.072 -0.070 -0.069 -0.068 

.45 -0.130 -0.123 -0.120 -0.118 -0.116 -0.115 -0.113 

.55 -0.178 -0.170 -0.167 -0.165 -0.162 -0.161 -0.159 

.65 -0.226 -0.219 -0.215 -0.213 -0.210 -0.209 -0.207 

.75 -0.278 -0.272 -0.269 -0.267 -0.264 -0.262 -0.259 

.85 -0.339 -0.338 -0.334 -0.331 -0.325 -0.322 -0.311 

64 
--------- -

Table 28. Normalized stress intensity factors calculated at the 

midsection of an internal planar elliptic crack with 

d/h::0.10 in a plate under uniform bending M. 

KA/Ko 

a/Lo_1.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lo/h Strain 

.05 0.216 0.218 0.219 0.219 0.220 0.220 0.221 

.15 0.263 0.270 0.273 0.274 0.276 0.276 0.279 

.25 0.308 0.321 0.325 0.328 0.331 0.332 0.337 

.35 0.357 0.376 0.383 0.387 0.393 0.395 0.403 

.115 0.1112 0.4110 0.451 0.457 0.467 0.471 0.11 84 

.55 0.1177 0.520 0.538 0.511 9 0.566 0.5711 0.598 

.65 0.571 0.6112 0.676 0.697 0.730 0.7116 0.7tJr 

.75 0.004 0.959 1.0115 1.101 1.199 1.251 1.11111 

KB/Ko 

a/Lo-1.0 2.0 3.0 11.0 7.0 10.0 Plane 

Lo/h Strain 

.05 0.170 0.172 0.173 0.173 0.174 0.1711 0.175 

.15 0.114 0.120 0.123 0.124 0:125 0.126 0.128 

.25 0.061 0.071 0.075 0.077 O.OBO 0.081 0.084 

.35 0.010 0.0211 0.030 0.033 0.037 0.038 0.043 

.115 -0.038 -0.020 -0.013 -0.009 -0.004 -0.001 0.005 

.55- -0.0811 -0.062 -0.053 -0.048 -0.0110 -0.037 -0.028 

.65 -0.128 -0.102 -0.089 -0.082 -0.071 -0.066 -0.052 

.75 -0.171 -0.137 -0.117 -0.105 -0.083 -0.073 -0.037 

65 
Table 29. Normalized stress intensity factors calculated at the 

midseotion of an internal planar elliptio crack with 

d/h::0.15 in a plate under uniform bending M. 

KAIKo 

a/Lo -.1.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lo/h Strain 

.05 0.313 0.317 0.319 0.319 0.320 0.321 0.323 

.15 0.355 0.367 0.371 0.373 0.376 0.378 0.382 

.25 0.399 0.420 0.428 0.432 0.438 0.440 0.449 

.35 0.450 0.482 0.495 0.502 0.512 0.517 0.532 

.45 0.514 0.563 0.584 0.596 0.614 0.623 0.650 

.55 0.610 0.689 0.727 0.750 0.786 0.004 0.860 

.65 0.846 1.015 1.106 1.167 1.270 1.325 1.524 

KB/Ko 

a/Lo _1.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lo/h StraJ.n 

.05 0.266 0.270 0.272 0.273 0.273 0.274 0.275 

.15 0.207 0.218 0.221 0.223 0.226 0.227 0.231 

.25 0.152 0.169 0.176 0.179 0.184 0.186 0.191 

.35 0.101 0.125 0.134 0.139 0.146 0.149 0.158 

.45 0.055 0.085 0.098 0.105 0.116 0.120 0.133 

.55 0.011 0.051 0.069 0.080 0.0S6 0.104 0.125 

.65 -0.028 0.028 0.058 0.077 0.108 0.123 0.176 

66 
Table 30. Normalized stress intensity factors calculated at the 

midsection of an internal planar elliptic crack with 

d/b::0.20 in a plate under uniform bending M. 

KAIKo 

a/Lo -1.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lo/h Strain I 

.05 0.410 0.417 0.419 0.420 0.421 0.422 0.425 

.15 0.445 0.464 0.471 0.474 0.479 0.481 0.488 

.25 0.490 0.522 0.534 0.541 0.551 0.555 0.569 

.35 0.549 0.600 0.621 0.633 0.651 0.660 0.687 

.45 0.642 0.725 0.764 0.787 0.823 0.841 0.897 

.55 0.875 1.050 1.142 1.202 1.303 1.357 1.549 

KBIKo 

a/Lo~1.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lo/h Strain 

.05 0.362 0.368 0.370 0.372 0.373 0.373 0.376 

.15 0.298 0.315 0.321 0.324 0.328 0.329 0.336 

.25 0.242 0.268 0.278 0.283 0.291 0.29J 0.304 

.35 0.193 0.229 0.244 0.252 0.264 0.269 0.285 

.45 0.148 0.197 0.219 0.232 0.252 0.261 0.289 

.55 0.109 0.181 0.218 0.242 0.280 0.299 0.365 

67 
Table 31. Normalized stress ~ntensi ty factors calculated at the 

midsection of an internal planar elliptic crack wi th 

d/h=0.25 in a plate under uniform bending M. 

KA/Ko 

a/Lo-71.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lo/h Strain 

.05 0.506 0.516 0.519 0.520 0.523 0.523 0.527 

.15 0.535 0.563 0.573 0.578 0.585 0.589 0.599 

.25 0.585 0.633 0.652 0.662 0.677 0.685 0.708 

.35 0.670 0.751 0.787 0.008 0.840 0.856 O. 907 

.45 0.895 1.066 1.152 1.206 1.298 1.346 1.515 

KB/Ko 

_ a/Lo-l.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lo/h Stral.n 

.05 0.457 0.466 0.469 0.471 0.473 0.473 0.477 

.15 0.388 0.412 0.421 0.425 0.431 0.434 0.443 

.25 0.332 0.370 0.385 0.393 0.405 0.410 0.427 

.35 0.285 0.340 0.363 0.377 0.398 0.407 0.438 

.45 0.305 0.332 0.369 0.395 0.437 0.459 0.513 

68 
· Table 32. Normalized stress intensity factors calculated at the 

midsection of an internal planar elliptic crack with 

d/h=0.30 in a plate under uniform bending M. 

KA/Ko 

a/Lo_l.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lo/h Strain 

.05 0.600 0.614 0.618 0.621 0.624 0.625 0.630 

.15 0.627 0.666 0.680 0.688 0.698 0.703 0.720 

.25 0.698 0.771 0.801 0.817 0.842 0.855 0.895 

.35 0.910 1.067 1.141 1.187 1.261 1.300 1.432 

KB/Ko 

a/Lo--l.0 2.0 3.0 4.0 7.0 10.0 Plane 

Lo/h Strain 

.05 0.550 0.563 0.567 0.570 0.572 0.574 . 0.579 

.15 0.477 0.511 0.524 0.530 0.539 0.543 0.557 

.25 0.423 0.479 0.501 0.514 0.533 0.542 0.570 

.~5 0.381 0.471 0.513 0.538 0.579 0.600 0.669 

Table 33. Normalized stress intensity factors calculated at the 

midsection of two opposite planar elliptic edge cracks 

in a plate under uniform tension N. 

a/Lo_l.0 2.0 3.0 4.0 10.0 100.0 Plane 

Lo/h Strain 

0.1 1.060 1.089 1.099 1.104 1. 113 1.119 1.123 

0.2 1.009 1.062 1.081 1.091 1.109 1.121 1.125 

0.3 0.966 1.028 1.065 1.079 1.106 1.124 1.129 

0.4 0.929 1.019 1.053 1.072 1.108 1. 131 1.137 

0.5 0.902 1.008 1.050 1.073 1.118 1.148 1.155 

0.6 0.902 1.038 1.080 1. 108 1. 165 1.203 1.212 

0.7 0.929 1.082 1.149 1.186 1.262 1. 315 1.325 

0.8 0.997 1. 195 1.284 1.336 1.445 1.524 1.539 

69 
Table 34. Maximum normalized stress intenai ty factors for 

two planar elliptic edge cracks in a plate 

under pure tension N. 

bla -- 0.1 1.0 4.0 20.0 Single 

Lo/h alLo Crack 

0.1 2 0.996 0.986 0.982 0.981 0.981 

0.1 4 1.072 1.065 1.062 1.062 1.062 

0.1 10 1.127 1.122 1.121 1.120 1.120 

0.2 2 0.979 0.959 0.951 0.949 0.949 

0.2 4 1.107 1.090 1.084 1.082 1.082 

0.2 10 1.221 1.210 1.207 1.206 1.206 

0.3 2 1.007 0.975 0.963 0.961 0.961 

0.3 4 1.189 1. 160 1.151 1.149 1.149 

0.3 10 1.382 1.364 1.359 1.358 1.358 

0.4 2 1.051 1.006 0.991 0.989 0.989 

0.4 4 1.299 1.255 1.243 1.240 1.240 

0.4 10 1.600 1. 571 1.563 1.562 1. :62 

0.5 2 1.106 1.048 1.032 1.030 1.030 

0.5 4 1.430 1.370 '1. 354 1.352 1.352 

0.5 10 1.879 1.836 1.824 1.822 1.821 

0.6 2 1.230 1.0 S6 1.080 1.077 1.077 

0.6 4 1.568 1.491 1.472 1.469 1.469 

0.6 10 2.208 2.141 2.124 2.121 2.121 

0.7 2 1.370 1.120 1.100 1.091 1.097 

0.7 4 1.694 1.575 1.553 1.550 '.550 

Ch7 10 2.532 2.434 2.409 2.405 2.405 

70 
Table 35. Maximum normalized stress intena1 ty factors for 

two planar elliptic edge cracks in a plate 

under uniform bending M. 

b/a -0.1 1.0 4.0 20.0 Single 

Lo/h a/Lo Crack 

0.1 2 0.874 0.864 0.861 0.860 0.860 

0.1 4 0.943 0.936 0.934 0.933 0.933 

0.1 10 0.992 0.988 0.987 0.986 0.9al 

0.2 2 0.766 0.728 0.719 0.718 0.718 

0.2 4 0.847 0.830 0.825 0.824 0.824 

0.2 10 0.940 0.930 0.927 0.927 0.927 

0.3 2 0.751 0.665 0.651 0.650 0.650 

0.3 4 0.803 0.755 0.745 0.744 0.744 

0.3 10 0.915 0.900 0.896 0.896 0.895 

0.4 2 0.792 0.677 0.659 0.658 0.656 

0.4 4 0.801 0.726 0.713 0.711 0.711. 

0.4 10 0.923 0.9)2 0.896 0.895 0.895 

- 0.5 2 0.826 0.684 0.663 0.661 0.659 

0.5 4 0.834 0.719 0.703 0.701 0.700 

0.5 10 0.950 0.910 0.902 0.901 0.901 

0.6 2 0.855 0.686 0.662 0.660 0.659 

0.6 4 0.909 0.743 0.724 0.722 0.721 

0.6 10 0.995 0.925 0.912 0.910 0.910 

0.7 2 0.874 0.683 0.658 0.655 0.654 

0.7 4 0.989 0.784 0.761 0.759 0.757 

0.7 10 1.064 0.956 0.939 0.937 0.936 

71 
Table 36. Normalized stress intensity factors calculated at the 

midsection of symmetrically lomted(d=O) two identical 

planar internal elliptic cracks in a plate under 

uniform tension N. 

b/a --. 0.1 1.0 4.0 20.0 Single 

Lo/h a/Lo Crack 

0.1 2 0.977 0.976 0.976 0.976 0.967 

0.1 4 0.987 0.987 0.987 0.986 0.987 

0.1 10 0.994 0.993 0.993 0.993 0.993 

0.2 2 O. 975 0.972 0.971 0.971 0.971 

0.2 4 0.995 0.994 0.993 0.993 0.993 

0.2 10 1.008 1.007 1.007 1.007 1.007 

0.3 2 0.980 0.980 0.979 0.979 0.979 

0.3 4 1.015 1.013 1.012 1.012 1.012 

0.3 10 1.035 1.034 1.034 1.034 1.034 

0.4 2 1.007 1.002 1.000 1.000 1.000 

0.4 4 1.051 1.048 1.047 1.047 1.0 lt7 

0.4 10 1.080 1.078 1.078 1.078 1.078 

0.5 2 1.046 1.039 1.037 1.036 1.036 

0.5 4 1.106 1.102 1.101 1.101 1. 1 01 

0.5 10 1.147 1.145 1.145 1.1 ~5 1.145 

0.6 2 1.108 1.098 1.095 1.095 1.095 

0.6 4 1.190 1.185 1.183 1.183 1.183 

0.6 10 1.248 1.246 1.245 1.245 1.245 

0.7 2 1.205 1.192 1.188 1.187 1.187 

0.7 4 1.321 1.313 1. 311 1.310 1.310 

()-.7 10 1.407 1.404 1.403 1.403 1.403 

0.8 2 1.367 1.348 1.342 1.341 1.341 

0.8 4 1.541 1.529 1.526 1.525 1.525 

0.8 10 1.681 1.676 1.674 1.674 1.674 

0.9 2 1.703 1.672 1.662 1.661 1.661 

0.9 4 2.007 1.988 1.982 1. 981 1. 981 

0.9 10 2.285 2.275 2.272 2.272 

72 
Table 37. Normalized stress intensi ty factors calculated at the 

midsection of two identical planar internal elliptic 

cracks in a plate under uniform tension N. 

KAIKo 

bla _ 0.1 1.0 4.0 20.0 

d Lo/h . alLo 

0.1 0.1 2 0.957 0.821 0.981 0.979 

0.1 0.1 4 1.002 0.994 0.992 0.991 

0.1 0.1 10 1.000 0.999 0.999 0.999 

0.1 0.2 2 1.027 0.986 0.977 0.976 

0.1 0.2 4 1.005 1.002 1.001 1.001 

0.1 0.2 10 1.018 1.018 1.018 1.018 

0.1 0.3 2 1.012 0.995 0.990 0.989 

0.1 0.3 4 1.034 1.032 1.031 1.031 

0.1 0.3 10 

0.1 0.35 2 1.021 1.010 1.005 1.004 

0.1 0.35 4 1.058 1.056 1.055 1.055 

0.1 0.35 10 

0.2 0.1 2 0.947 0.783 0.979 0.976 

0.2 0.1 4 1.006 0.995 0.992 0.992 

0.2 0.1 10 1.003 1.002 1.002 1.002 

0.2 0.2 2 1.052 0.993 0.981 O. gf9 

0.2 0.2 4 1.020 1.016 1.014 1.014 

0.2 0.2 10 1.039 1.038 1.038 1.038 

- 0.2 0.25 2 1.056 1.003 0.994 0.992 

0.2 0.25 4 1.044 1.040 1.040 1.039 

0.2 0.25 10 1.073 1.073 1.073 1.073 

0.3 0.1 2 0.933 0.760 0.980 0.975 

0.3 0.1 4 1.018 1.003 0.999 0.998 

0.3 0.1 10 1.014 1. 013 1.013 1.013 

0.3 0.15 2 0.814 0.985 0.988 0.985 

0.3 0.15 4 1.033 1.024 1.022 1.021 

0.3 0.15 10 1.047 1.046 1.046 1.046 

73 
Table 37- Cont. 

KB/Ko 

b/a -a- 0.1 1.0 4.0 20.0 

d Lo/h a/Lo 

0.1 0.1 2 0.959 0.830 0.982 o. 1Jl9 

0.1 0.1 4 1.002 0.993 0.991 0.991 

0.1 0.1 10 0.999 0.999 0.999 0.999 

0.1 0.2 2 1.020 0.984 0.978 0.976 

0.1 0.2 4 1.002 1.000 0.999 0.999 

0.1 0.2 10 1.014 1.014 1.014 1.014 

0.1 0.3 2 1.004 0.989 0.985 0.984 

0.1 0.3 4 1.021 1.020 1.019 1.019 

0.1 0.3 10 

0.1 0.35 2 1.006 0.91Jl 0.993 0.993 

0.1 0.35 4 1.036 1.035 1.034 1.034 

0.1 0.35 10 

0.2 0.1 2 0.950 0.799 0.1Jl9 o. CJ76 

0.2 0.1 4 1.004 0.994 0.992 0.991 

0.2 O. 1 10 1.001 1.001 1.001 1.001 

0.2 0.2 2 1.038 0.987 0.977 0.976 

0.2 0.2 4 1.010 1.007 1.006 1.005 

0.2 0.2 10 1.026 1.026 , .026 1.026 

0.2 0.25 2 1.034 0.990 0.983 0.982 

0.2 0.25 4 1.023 1.021 1.020 1.019 

0.2 0.25 10 1.046 1.046 1.046 '.046 

0.3 0.1 2 0.935 0.778 0.978 0.974 

0.3 0.1 4 1. 013 0.999 0.9£6 0.995 

0.3 0.1 10 1.009 1.00g 1.008 1.008 

0.3 0.15 2 o. B2 9 o. CJ77 0.979 0.977 

0.3 0.15 4 1.019 1.011 1.008 1.008 

0.3 0.15 10 1.030 1.030 1.030 1.030 
Table 38. Normalized stress intensi ty factors calculated at 

the midsection of three identical planar internal 

elliptic cracks in a plate tmder tmiform tension N. 

KAlKa for the middle crack 

b/a- 0.1 1.0 4.0 20.0 

d Lolh alLo 

0.1 0.1 2 0.983 0.981 0.979 0.979 

0.1 0.1 4 0.994 0.992 0.992 0.991 

0.1 0.2 2 0.985 0.900 0.977 0.976 

0.1 0.2 4 1.007 1.004 1.002 1.001 

0.1 0.3 2 1.006 0.997 0.991 0.989 

0.1 0.3 4 1.040 1.035 1.032 1.031 

0.2 0.1 2 0.981 0.978 0.976 0.975 

0.2 0.1 4 0.995 0.994 0.992 0.992 

0.2 0.2 2 .0.993 0.986 0.981 0.979 

0.2 0.2 4 1.023 1.018 1.015 1.014 

0.3 0.1 2 0.983 0.979 0.976 0.975 

0.3 0.1 4 1.003 1.001 0.999 0.998 

IB/Io for the middle crack 

0.1 0.1 2 0.984 0.981 0.980 0.979 

0.1 0.1 4 0.993 0.992 0.991 0.991 

0.1 0.2 2 0.984 0.980 0.977 0.976 

0.1 0.2 4 1.003 1.001 0.999 0.999 

0.1 0.3 2 0.997 0.990 0.9e6 0.984 

0.1 0.3 4 1.026 1.022 1.020 1.019 

0.2 0.1 2 0.981 0.978 o. ':!T7 0.976 

0.2 0.1 4 0.994 0.993 0.992 0.991 

0.2 0.2 2 0.9f57 0.981 o. fJl7 0.976 

0.2 0.2 4 1.012 1.009 1.006 1.005 

0.3 0.1 2 0.981 0.5'77 0.975 0.974 

0.3 0.1 4 0.999 0.997 0.996 0.995 

75 
Table 38-Cont. 

KAIKo for outer crack 

b/a -- 0.1 1.0 4.0 20.0 

d Lo/h alLo 

0.1 0.1 2 0.982 0.980 0.979 0.979 

0.1 0.1 4 0.993 0.992 0.991 0.991 

0.1 0.2 2 0.982 0.978 0.976 0.976 

0.1 0.2 4 1.005 1.003 1.002 1.001 

0.1 0.3 2 0.999 0.994 0.991 0.989 

0.1 0.3 4 1.037 1.034 1.032 1.031 

0.2 0.1 2 0.979 0.977. 0.976 0.975 

0.2 0.1 4 0.994 0.993 0.992 0.992 

0.2 0.2 2 0.988 0.983 0.980 0.979 

0.2 0.2 4 1. 019 1. 016 1.014 1.014 

0.3 0.1 2 0.980 0.978 0.976 0.975 

0.3 0.1 4 1. 001 1.000 0.999 0.998 

KB/Ko for outer crack 

0.1 0.1 2 0.982 0.980 0.980 0.979 

0.1 0.1 4 0.993 0.992 0.991 0.991 

0.1 0.2 2 0.981 0.978 0.976 0.976 

0.1 0.2 4 1.002 1.000 0.999 0.999 

0.1 0.3 2 0.992 0.988 0.985 0.984 

0.1 0.3 4 1.023 1.021 1.020 1.019 

0.2 0.1 2 0.979 0.978 0.976 0.976 

0.2 0.1 4 0.993 0.992 0.991 0.991 

0.2 0.2 2 0.983 0.979 0.977 0.976 

0.2 0.2 4 1.010 1.007 1.006 1.005 

0.3 0.1 2 0.978 0.976 0.974 0.974 

0.3 0.1 4 0.998 0.995 0.995 0.995 

76 
Table 39. NormallZed stress intensity factors on the crack front 

for an internal elliptic crack with d/h=0.20, Lih=0.45, 

a/Lo=4. 

x (KA/Ko)N (KB/Ko)N (KB/Ko)M (KB/Ko)M 

0.99 0.426 0.411 0.186 0.151 

0.90 0.723 0.707 0.357 0.218 

0.80 0.865 0.831 0.454 0.229 

0.70 O. gr4 0.916 0.531 0.233 

0.60 1.065 0.980 0.595 0.233 

0.50 1.143 1.028 0.649 0.233 

0.40 1.211 1.068 0.697 0.233 

0.30 1.266 1.096 0.735 0.233 

0.20 1.307 1.116 0.763 0.233 

0.10 1.333 1.128 0.781 0.233 

0.00 1.342 1.132 0.787 0.232 

77 
Table 40. Normalized stress intensity factors on the crack front 

for an internal elliptic crack with d/h=0.10, Lo/h=0.lJ5, 

a/Lo=4. 

x (KA/Ko)N (KB/Ko)N (KB/Ko)M CKB/Ko)M 

0.99 0.411 0.393 0.097 0.065 

0.90 0.692 0.686 0.205 0.072 

0.80 0.816 0.804 0.269 0.057 

0.70 0.902 0.883 0.318 0.041 

0.60 0.967 0.941 0.357 0.028 

0.50 1.018 0.984 0.388 0.016 

0.40 1.061 1.019 0.414 0.007 

0.30 1.092 1.044 0.lJ33 0.000 

0.20 1.114 1.061 0.446 -0.005 

0.10 1.128 1.072 0.455 -0.008 

0.00 1.132 1.075 0.l!57 -0.009 

78 
Table 41. Normalized stress intensi ty factors on the crack front 

for an elliptic edge crack with Lc/h=0.45, a/Lo=4. 

x (KN/Ko) (KM/Ko ) 

0.99 0.762 0.646 

O.go 0.920 0.696 

0.80 1.027 0.711 

0.70 1.084 0.701 

0.60 1.130 0.695 

0.50 1.166 0.689 

0.40 1.197 0.686 

0.30 1.214 0.680 

0.20 1.227 0.676 

0.10 1.237 0.675 

0.00 1.240 0.675 

79 
FIG U RES 

80 
-Q 

/ 

X2 

F~gure 1. 

-+---x, (a) 

a 

(b) 

(c) 

-+. ....... .", ~~ .... /-+-X_2)-- N (X,) 

(d) M(Xd 

NotatJ.on for the part-through crack problem. 

81 
L 8 h N ---+--+--- -r.--+--- N ,-

(0) 

N 

N 

a 8 8+d8 

(b) 

Figure 2. Notation for the related plane strain probleL 

82 
N 

_ ~ __ LG ... A--,..d N ----- - -~-+--

M __ ~ ____________ 81 

~ ______________ 8 ~M 

(b) 

Figure 3. Geometry and notation for an internal crack. 

83 
~I 2 

I a 

Figure 4. 

L 

2 

Geometry and notation for two opposite elliptic 

edge cracks. 

84 
t 

X3 

I 

I c 

~ b 

2a I 

\ I , 

) _LK -+ fJ;t: :.::f=:-E _+-~_-L - ) 

/- J 

( h I t ~ x, 

I 

I 

Figure 5. Geometry and notation for two identical cracks. 

Figure 6. Geometry and notation for three identical cracks. 

85 
1.E 

1.6 

1.4 

1.2 

1.0 

0.05 0.1 0.15 0.2 0.25 0.3 d/h 

1 : Lo/h=O.65, a/Lo=7, 

2: L/h=O. 65, a/Lo=3, 

3: Lo/h=O.55, a/Lo=7, 

4: L/h=O. 55, a/Lo=3, 

5: L/h=O. 45. a/Lo=7. 

6: Lo/h=O. 45. a/Lo=3, 

7: Lo/h=O. 35, a/Lo=7, 

8: Lo/h=O. 35, alLo=3, 

9: Lo/h=O.25, a/Lo=3. 

Figure 7. Effect of d/h ratio on the normalized stress intensi ty 

factors calculated at the midsection of var~ous 

internal elliptic cracks under pure tension. 

86 
1.4 

6 

1.3 

2 

1.2 / 

1 

1.1 

9 

-1.0 ---------

0.05 0.1 015 0.2 0.25 0.3 Cl/h 

1 : Lo/h=O. 35, a/Lo=3, 

2: Lo/h=O. 35, a/Lo=7, 

3: Lih=O.65, a/Lo=3, 

4: Lo/h:O. 55, a/Lo=3, 

5: Lo/h=O. 45, a/Lo=3, 

6: Lo/h:O. 45, a/Lo=7, 

7: Lo/h=O.55, a/Lo=7, 

8: Lol h:O. 65, a/Lo=7, 

9: L o/h=O.25, a/Lo=3. 

Figure 8. Effect of d/h ratio on the normalized stress intensity 

factors calculated at the midsection of various 

internal ell~ptic cracks under pure tension. 

87 
1.1 

0.9 

07 

o 5 

0.3 

-

0·05 0.1 0.15 02 025 0.3 d/h 

1 : L o/h=O.25, a/Lo=3, 

2: Lo/h=O.25, a/Lo:7, 

3: Lo/h=O.35, a/Lo=3, 

4: Lo/h=O. 35, a/Lo=7, 

5: Lo/h=0.45, a/Lo=3, 

6: Lol h=O. 45, alLo=7, 

7: Lo/h=0.55, a/Lo=3, 

8: Lol h=0. 55, a/Lo=7, 

9: L o/h=O.65, a/Lo=3, 

10: L o/h=0.65, a/Lo=7, 

11 : Lo/h=O. 15, a/Lo=3, 

12: Lo/h=0.15, a/Lo=7. 

Figure 9. Effect of d/h ratio on the normalized stress intensi ty 

factors calculated at the midsect10n of various 

1nternal ellipbc cracks under pure bending. 

88 
Ka/Ko 

0·6 

0.4 

0.2 

0.0 

- 0.2 

005 

1 : 

2: 

3: 

4: 

5: 

6: 

0.1 0.15 0.2 0.25 0.3 d/h 

Lo/h=O.15, a/Lo=3, 

Lo/h=O.25, a/Lo=3, 

Lo/h=O. 35, alLo=3, 

Lo/h=O.45, a/Lo=3, 

Lo/h=O. 55, a/Lo=3, 

Lo/h=O.65, a/Lo=3, 

Figure 10. Effect of d/h ratio on the normalized stress intensi ty 

factors calculated at the midsection of various 

internal elliptic cracks under pure bending. 

89 
Figure 11. 

0.05 

0.1 

0.15 

2.2 

0.2 

025 

'·8 

d / h = 0.3 

1 4 

1.0 

0.05 0.25 0.45 0.65 

Effect of Lo/h ratio on the normalized stress 

intensity factors calculated at the midsection of 

internal ell~ptic cracks with a/Lo=4 and 

different d/h ra tios under pure tension. 

90 
KS/Ko 

Figure 12. 

1.4 

12 -

005 0-25 

Effect of Lo/h ratio on the normalized stress 

intensity factors calculated at the midsection of 

internal elliptic cracks with a/Lo=4 and 

different d/h ra tios under pure tension. 

91 
KAlKa 

Figure 13. 

1.0 

07 

0·4 

0.1 

005 0.25 0.45 0.65 

Effect of Lo/h ratio on the normalized stress 

intensity factors calculated at the ml.dsection of 

internal ell~ptic cracks with a/Lo=4 and 

different d/h ratios under pure bending. 

92 
KS/Ko 

Figure 14. 

0.5 

0.2 

- 0.1 

- 04 

~/h=O.3 

~0.25 

~_O.2 

~-O.15 

1--,,..---,.--.----.----.--...... - ~ --.----... -~) 

0.05 0.25 0.45 065 O.85 L Ih 

o 

Effect of Lo/h ratio on the normalized stress 

intensity factors calculated at the midsection of 

internal elliptic cracks with a/Lo=4 and 

different d/h ra tios under pure bending. 

93 
1.3 

1.1 

0.9 

0.7 

05 

0.3 

Figure 15. 

- -- ---- -- --- --~--

.55 

~ .45 

~ .35 

.,,---- .25 

~ .15 

.05 

2 4 6 8 10 

a/Lo 

Effect of a/Lo ra~o on the normalized stress 

intensity factors calculated at the midsection of 

internal elliptic cracks with d/h=O.15 and 

different Lo/h ratios under pure bend~ng. 

94 
KS/ Ko 

0.22-

Figure 16. 

_------------.05 

_----------.15 

--- __ --------.25 

_--------.35 

_---::===---==--.45 ----------~ .55 

) 

3 5 7 9 a/lo 

Effect of a/Lo ratio on the normalized stress 

intensity fact¢rs calculated at the ~dsect1on of 

internal ell~pt1c cracks with d/h=O.15 and 

different Lo/h ratios under pure bending. 

95 
2·4 

2.2 

1.8 j 

1.4 

1.0-

Figure 17. 

Lo / h = .65 

.55 

.45 

.35 

~ 

t. .15 .25 

1 3 5 7 9 

Effect of a/Lo ratio on the normalized stress 

intensity factors calculated at the midsection of 

internal elliptic cracks with d/h=O.15 and 

different Lo/h ra tios under pure tension. 
KS/Ko lo I h = .65 

1.3 

.55 

-: 

.45 

1.1 J 

. .35 I 

i .25 

'--, 

1 .05 .15 

. 

I 

I 0.91 

I 

, 

I 

I 

1 , I :> 

1 3 5 7 9 a/lo 

Figure 18. Effect of alLo ratio on the normalized stress 

intensity factors calculated at the midsection of 

internal ell~ptic cracks with d/h=O.15 and 

different Lo/h ra tios under pure tension. 

97 
K/Ko 

1.7 

1 5 

1.3 

1.1 

09 

0.7 

Figure 19. 

1 

"" /' --- ------ -- -

---- ---- ----- -- . 5 --

_ - - - - - - - -=-=-= - - - - - .2 

-

3 5 

Line spring 

----- Ref. [19] 

7 9 

Comparison of the normall.Zed stress intensl.ty 

factors calculated l.n thl.s study at x=O for an 

internal planar elliptl.c crack in a plate under 

unl.form tension and wl.th the cor res t:0nding 

values given in Ref. [19]. 

98 
K/Ko 

1 8 

1.6 

1 4 

1.0 

Figure 20. 

I 

.J , 

------

0.1 0.3 0·5 

----- Line spring 

- - - - - - Ref. [1 9] 

a/Lo=10 

0.7 

2 

/ 

Lo/h 

Comparison of the normalized stress intens~ty 

factors calculated in th~s study at x=O for an 

internal planar elliptic crack ~n a plate under 

uniform t;enSl.on and with the correspond~ng 

values giYen in Ref. [19]. 

99 
K/Ko 

La/h = .9 

0.7 

05 -----.8 

-----.7 

0.3 

-----.6 

_-----------==.;=-==---- .5 --

_--------------- - .4 

.3 

0·' .2 

--_._----------------- .1 

1 3 5 7 9 a/La 

Figure 21. Normalized stress intensi ty rators calculated at 

the midsection of a symmetr~cally located (d/h=O) 

~nternal planar ell~ptic crack in a plate under 

uniform bending. 

100 
--- ~- - --- -- ---~ _ .. ~------- -"--

-K~~1 0.9 

1 

I 

I 1.2

1 

1.0 lL-,..--r-, _=_0_. S-,---.-_...,..., -r----..----r-..... , --» 

2 4 b 8 10 a/La 

Figure 22. 

1 0 

Normalized stress intensity factors calculated at 

the midsection of two opposite planar elliptic 

edge cracks ih a plate under uniform tension. 

line spring 

Ref. [20] 

0.8 v 

4 0--' 60'" 

Figure 23. Comparison of the stress intensity factors calculated 

in this study at the midsection of a symmetrically 

located internal(d/h=O) planar elliptic crack in a 

plate under un~form tension with the corresponding 

results given in [20].L Ih=.57, aIL =1.25, x=cos 8. o 0 

101 
REFERENCES 

1. F. Delale and F. Erdogan, "Line Spring Model for Surface 

Cracks in a Reissner Plate", Int. J. Eng. Sci., Vol. 1 9, 

1981, pp. 1331 • 

2. F. Delale and F. Erdogan, "Transverse Shear Effect in a 

Circumferentially Cracked Cylindrical ~ell", Quart. 

Appl. Math., Vol. 37, 1979, pp.239 • 

3. J. R. Rice and N. Levy, "The Part-'Ihrough Surface crack 

in an Elastic Plate", J. of Appl. Mech., Trans. AS ME , 

Vol. 39, 1 97 2, pp. 1 85 • 

4. F. Erdogan, "Mixed BoundarY-Value Problems in Hechanics: 

Addendum", Mechanics Today, S. Nemat-Nasser(ed), Vol. 4, 

pp. 199 • 

5. F. W. Smith and D. R. Sorensen, "The Semi-Elliptical 

Surface Crack - A Solution by the Alternating Mathod", 

Int. J. of Fracture, Vol. 12, 1976, pp.47 • 

6. R. C. Shah and A. S. Kobayashi, "On the Surface Flaw 

Problem", The Surface Crack: Physical Problems and 

Camp uta tional Sol utJ.ons, ed. J. L. Swedlow, 1972, pp.79 • 

7. J. Heliot, R. C. Labbens, A. Pellisier-Iannon, 

Elliptic Cracks in a Cylinder SubJected to 

Gradients", Fracture Mechanics, ASTM, SIP 677, 

pp.341 • 

"Semi 

Stress 

19'{9, 

8. I. S. Raju, J. C. Newman, Jr., "Stress-Intensity Factors 

for a W~de Range of Semi-Elliptl.cal Surface Cracks in 

Fini te-Thickness Plates", J. of Eng. Fracture Mechanics, 

Vol. 11, 1 g'{ 9, pp. 817 • 

9. J. C. Newman, Jr., "A Review and Assessment of the 

Stress-Intensity Factors for Surface Cracks", NASA 

Techmcal Memorandum 78&15, Nov. 1978. 

10. S. N. Atluri, Ie. Ka thiresan, A. S. Kobayash~, 

M. Nakayaki, "Inner Surface Cracks in an Internally 

Pressurized Cylinder Analyzed by a Three-Dimensional 

Displacement-Hybr~d Finite Element Mathod", ASHE, Part 

III, New York, 1977, pp.527 • 

102 
11. F. Delale and F. Erdogan, "The Effect of Transverse Shear 

in a Cracked Pla te Under Skewsymmetric Loading", J. Appl. 

Mech., Trans. ASHE, Vol.46 • 

12. D. M. Parks, "'!he Inelastic Line Spring: Estimates of 

Elastic-Plastic Fracture Parameters for Surface Cracked 

Plates and Stells", ASME, Paper 80-C2/PVP-109, 1980. 

13. D. M. Parks, "Inelastic Analysis of Surface Flaws Using 

the Line Spring M:ldel", Proceedings of the 5'th Int. 

Conf. on Fracture, Cannes, France, 1981. 

14. F. Delale and F. Erdogan, "Application of the Line Spring 

Model to a Cylindrical Shell Containing a Circumferential 

or an Axial Part-Through Crack", J. Appl. Mech., 1rans. 

ASHE, Vol. 4 9, 1982, pp.97 • 

15. F. Erdogan and IL Boduroglu, "Surface Cracks in a Plate 

of Finite Width Under Extension or Bending", Theoretical 

and Applied Fracture MechaniCS, VOl.1, 1985(to appear). 

16. F. Erdogan and O. S. Yahsi, "A Cylindrical Shell With a 

Stress-Free End Which Contains an Axial Part-'!hrough or 

Through Crack", Project Report, NASA NGR 39-007-011., 

Lehigh University, Dec. 1983. 

1'1'. F. Erdogan and O. S. Yahsi, "A Pressurized Cylindrical 

Shell With a Fixed End Which Contains an Axial PartThrough or Through Crack", Project Report, NASA NGR 

39-007-011, Leh~gh University, Dec. 1983. 

18. F. Erdogan and G. D. Gupta, "The Problem of Edge Cracks 

in an Infinite Strip", J. Appl. Mechanics, Trans. ASHE, 

1 97 4, pp. 1 001 • 

19. J. C. Newman, Jr. and I. S. Raju, "Stress Intensity 

Factor Equa tions for Cracks in Three-Dimensional Finite 

Bodies", ASTM STP-791, 1983, pp. 238 • 

20. T. Nishioka and S. N. Atluri, "Analytical Solution for 

Embedded Elliptic Cracks, and Fini te Element Al terna ting 

~thod for Elliptical Surface Cracks, SubJected to 

Arbi trary Loadings", Engineering Fracture MechaniCS, 

Vol. 17, 1983, pp.247 • 

103 
APPENDIX I 

'!he dimensionless q uan ti ties used in the formula t1on: 

X=-

u = - , 

2 112 

p = [ ] 

Ie ( 1-.) 

ai(1 - ,2) 

,R1 = 21rhA 4 

41( (1 - p) 

(1 + p) 

104 

w = 

Ie = 

5 (1 - 10') a2 

1 

3 + ... 

R2 = ---

, + ... 

4 

1 + ., 
APPENDIX II 

Procedure for finding coefficients of the shape functions Cgt 

and gb) are as follows. Loading should be seperated as pure tension 

and pure be nding. For pure tensi on case 

m = 0 

By using (8), (9) and (10a) 

A. 

K(s) = is V~1r-L-(x-) L b (s)2(i-1) 

1 i.w\ i 

K(s) 

Values of the left handside can easily be found for a wide range of 

s values in related literature, which enables us to create n 

equations Wl.th bi ' s as the n unknowns. Solution of this system is 

the coefficient of corresponding shape function. 

Same procedure l.S followed under pure bending to fl.nd ci ' s • 

105 
----,- ~--- ---~--~--------~-

APPENDIX III 

Solution or the Integral Equations 

i- Gauss-Chebyshev Closed Type Integration Formula 

'!he solution or the integral equa. tion which only has the Cauchy 

type singularity 

~f~et) dt + Jlkex• t) +et) dt = 

11' t-x 

-I -I 

may be expressed as 

1 1 

K(x, t) = - -+ k(x, t) 

4ft" t-x 

rex) -1 < x < 1 

- 11' 1 1\.-\ 1 

- [- K(xj,t1 )g(t1) + I K(xj,ti)g(ti ) + - K(xj,tn)g(tn ) ] 

n-1 2 i..~ 2 

j = 1, ••••• , ( n-1 ) 

ti = cos [.:: 11'] 

n-1 

i=1, ••..•• ,n 

106 
- ------ - - -- --.~~------------

x J = co s [ 2 j-1 .. ] 

2n-2 

and the additional. condition 

becomes 

I I t(t) dt = A , 

-\ 

'If 1 "'-_I 1 

j = 1, •••••• , (n-1 ) 

[-g(t 1) + I g(ti ) + - g(tn ) J = A 

n-1 2 1..~ 2 

One may find the detailed derivation of the formula in [4J. 

When there is a system of integral. aqua tions with more than one 

unknown functions to be sol ved, the above procedure can be used by 

applying the formula to each integra tion seperately. For example 

1 J k21 (x,t)·1(t) 

-I 

f1 (x) 

107 
1 J .1(t) dt = Al ' 

-I 

can be solved as 

1 J • 2 ( t) d t = A2 J 

-I 

-1 < x < 1 

s has ~en introduced to aef ine different colloca tJ.on points for 

each functions. 

1 1 

-+ k 11 (x,t) 

t-x 

1 1 

K22 (x,s) = - -+ k 22 (x,s) 

11' s-x 

108 
---- ---- ------------- ------ -~--- ---~--- --~--

1r 1 "",-I 1 

[- K11 (xj ,t1 )S,(t,) + LK11(Xj,t1)S,(t1) +-K11(Xj,tn1)S,(tn1)] 

n1-1 2 "'-a 2 

1r 1 "'2-1 

+-[-K12 (Xj ,S1)S2(S1) + LK12 (Xj ,S1)S2(Si) 

n2-1 2 ~ 

1 

+-K'2(Xj,sn2)S2(sn2) ] = f 2 (X j ) , 

2 

[ 

2j-1 ] 

x j = cos 11 

2n1-2 

j = 1, ••••• , (n1-1) 

1r 1 ft.-I 1 

[- K21 (xj ' t1 )S1 (t 1) +.L K21 (x j' t 1 )S1 (ti ) + -K21 (Xj, tn1 )S1 (tn1 )] 

~-1 2 ~a 2 

11' 1 1lz-' 

+-[- K22 (X j ,S1 )g2(s1) + I K22 (x j'S1)g2(Si) 

n2-1 2 "-.a. 

1f 

1 

+-~2(Xj,sn2)g2(sn2) ] = f 2 (X j ) 

2 

1 '!\t-I 1 

- [- S,(t 1) 

n,-1 2 

+ L S1 (t~) +- S, (tn1 ) ] 

2 .tea 

1f , ~_I 1 

[- S2(s,) + L S2(si) +- g2(sn2) ] 

~-1 2 .\..,. 2 

109 

j = " •••••• ,(n2-') 

= A, , 

= A2 ' 
i = 1, •••••• , n1 

[ 

i-1 ] si = cos _1r , 

~-1 

i = 1, •••••• ,~ 

Above solution can also be used for any number of unlalown functions 

in the same manner. 

ii- Order of Singularity in the Integral Equation 

At . first glance, it seems that (31) has singularities other 

than Cauchy type. But if we take a closer look to the eq t.a tion, we 

will see that modif~ed Bessel function of the second ldnd has the 

follOWing property for relatively small values of the argument. 

K (x) - 2 x-2 

2 

By using above property, we can say that 

2 1(1-p) 

lim K2 (plt-xl) = =--- (t-x)~O p2{t_x)2 {t_x)2 

110 
which gives 

[ 

4 1 41( ( 1-' ) 

lim - - K2 (plt-x I) - ------ (t-x)-O 1 ... t-x 1... (t-x)3 

1 

] = a 

This proves that (31) has only Cauchy type singularity and can be 

sol ved by Gauss-Chebyshev closed type integration formula. 

Integrals on the interval (-1 , x) can be written on (-1 , +1) 

by redifining the kernels by the use of a step function H(y) as 

follows: 

x J ={ ret) dt H(x-t) f(t) dt 

-I -I 

where, 

{ ~ if Y < a 

H(y) = ~f y ~ a • 

111 
APPENDIX IV 

Numerical Integration 

Gauss-Chebyshev closed type integrab.on f'ormula is used f'or 

numerical integration purposes. Any integr-al in the interval 

(-1 , +1) can be expressed as, 

I 

I f'(x) dx 

~1-x?-

-I 

If' an integration in the ~nterval (-1 , x) is desired, above 

f'ormula can again be used by mul tiplYJ.Ilg the integrand wi th the H 

function defined in Appendix III. 

Number of po~nts (n) in the integra tion should be much bJ..gher 

than the number of' points used in the solution of' integral equa tions 

if one desires to evaluate the stress intena~ ty f'actor or the crack 

f'ront at any point accurately. Values of the funct~ons to be 

integrated at any point can be found by interpola t~ng the f'uncbon 

between the two surrounding known val uea f'ound from the sol ut~on of' 

integral equa tiona. 

112 
1 Report No 2 Government Acce5Slon No 3 Recipient's Catalog No 

NASA CR-177926 

4 Title and Subtitle 5 Report DatI' 

Interaction of Part-Through Cracks in a Flat Plate I--Q,joAln.u..lr..L..ii 'J-...L. 11 qOL!oolR~!5 ____ ._ 

6 Performeng Organization CodE: 

7 Author(s) 8 Perform eng Organezeltlon Report No 

B. Aksel and F. Erdogan 

t---------------:---------------~ 10 Work Unit No 

9 Performing Organization Name and Address 

Lehigh University Department of Mechanical Engineering 

Bethlehem, PA 

11 Contract or Grant No 

NGR 39-007-011 

r-:--:-----------------------------l 13 Type of Report and Pereod Covered 

12 Sponsoreng Agency Name and Address 

National Aeronautics and Space Administration rnnt\,,;lrtn\" Rpnnrt 

Washington, DC 20546 14 Sponsoring Agency Code 

505-33-23-02 

15 Supplementary Notes 

Langley Technical Monitor: W. Steven Johnson 

16 Abstract 

Main purpose of this study is to determine the accuracy of the line spring 

model, to investiqate the effect of interaction between two and three cracks, and 

to provide extensive numerical results which may be useful in app1icati~ns. Line 

spring model with Reissnerls plate theory is formulated to be used for any number 

and configurations of cracks provided that there is symmetry. This model is used 

to find stress intensity factors for elliptic internal cracks, elliptic edge cracks 

and two opposite elliptic edge cracks. Unfortunately, because of the unavailability of previous work done on the cases considered, only stress intensity factors for 

central elliptic crack could be compared with other methods. Despite the simplicity 

of the line spring model, the results are found to be surprisingly close. 

17 Key Words (Suggested by Authorlsl) 

line spring model, cracks, 

Reissnerls plate theory, 

stress intensity factors 

19 S'!'Curlry Oasslf (of this report) 

Unclassified 

• 

18 Distribution Statement 

Unclassified - Unlimited 

Subject Category 39 

20 Security Classl' (of this page) 

Unclassified 

21. No of Pages 

113 

For sale bV tho Natlona' TechnlCsl Information Senllce, Springfield, Virginia 22151 

A06 
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