THE SURFACE CRACK PROBLEM IN AN 

ORTHOTROPIC PLATE UNDER BENDING AND TENSION 

BING-HUA WU and F. ERDOGAN 

(NASA-CR-179992) THE SIIhFACE CRACK PHOBLEW Nt3 7- 12020 

IN AN OElHOIhOPIC FLA3E UHCEK ELKDING AND 

TENSICN (Lehiqh Uriiv.) 78 F CSCL 20K 

Unclas 

G3/39 44911 

November 1986 

Lehigh University 

Bethlehem, PA 

THE NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GRANT NO. NGR-39-007-011 
THE SURFACE CRACK PROBLEM IN AN 

ORTHOTROPIC PLATE UNDER BENDING AND TENSION 

bY 

BING-HUA WU and F. ERDOGAN 

November 1986 

Lehigh University 

Bethlehem, PA 

THE NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GRANT NO. NGR-39-007-011 
Abstract 

In this study the elasticity problem for an infinite orthotropic flat plate 

containing a series of through and part-through cracks and subjected to bending 

and tension loads is considered. The problem is formulated by using Reissner’s 

plate bending theory and considering three dimensional material orthotropy. The 

Line-spring model developed by Rice and Levy is used to formulate the surface 

crack problem in which a total of nine material constants have been used. The 

main purpose of this study is to determine the effect of material orthotropy on 

the stress intensity factors, to investigate the interaction between two 

asymmetrically arranged collinear cracks, and to provide extensive numerical 

results regarding the stress intensity factors. The problem is reduced to a 

system of singular integral equations which is solved by using the GaussChebyshev quadrature formulas. The calculated results show that the material 

orthotropy does have a significant effect on the stress intensity factor . 

I 

1 
Chapter 1 

Introduction 

In recent years the increasing use of composite plates in many engineering 

structures and especially in the aero-space industry has brought up the need for 

more intensive stress analysis of anisotropic materials. Up to now no completely 

analytical solution of the related failure problems has been available due to the 

inherent difficulties involving the stress analysis and material characterization. In 

many cases the failure is attributed to the growth of cracks or crack-like flaws 

which exist in the structure. During the past two decades many investigators 

have studied the stress state in the immediate neighborhood of the crack tip 

since the local fracture of a structure appears to be governed mainly by this 

stress field. The singular behavior of the stress state near the crack-tip as 

characterized by the stress intensity factor depends on the magnitude of the 

external loads, the configuration of the body including the crack size and shape, 

and the material properties. Problems of a surface crack in a structural 

component which may locally be represented by "plate" or "shell" have long 

attracted the attention of investigators who have been interested in fracture 

mechanics. In the past few years there has been some renewed interest in the 

Line-spring modei, (which was originally devejoped by Rice "" LlJJ r-- 1Ul DUlVlll& --'-*:-- 

surface crack problems in plates and shells. This interest seems to be justified 

because of the relatively high accuracy of the results obtained from this model 

(see, for example, [2], [3] and [4] ). The purpose of this study is to extend the 

method to orthotropic elastic plates. The stress intensity factors for two 

orthotropic plates and one isotropic plate are extensively studied for through 

and part-through cracks. Here the isotropic case has been considered for the 

2 

. 
purpose of comparison. In materials undergoing fracture it is often observed 

that small pre-existing cracks or cracks generated during service join together 

and grow into major cracks to cause final failure. Thus, theoretical analysis of 

the interaction of multiple cracks in the structure is quite important. 

Consequently, the emphasis in this study will be on the interaction of two 

collinear, semi-elliptic surface cracks or through cracks in an infinite plate 

subjected to uniform bending or tension. This problem is formulated by using 

Reissner’s plate bending theory and considering the material orthotropy with the 

isotropic medium as a special case. The problem is reduced to a system of 

singular integral equations. The effect of material orthotropy on the stress 

intensity factor and the interaction between two arbitrarily arranged collinear 

cracks have been investigated. Rather extensive numerical results have been 

provided which may be useful in application. 

3 
Chapter 2 

The general formulation of the problem 

2.1 The governing equations for an orthotropic plate under 

bending and tension 

In this section a brief derivation of the fundamental equations for an 

orthotropic plate under bending and in-plane stretching is given. The problem of 

bending and the problem of in-plane loading are uncoupled. The material of the 

plate is assumed to be orthotropic with principal axes or orthotropy parallel to 

the co-ordinate axes. Thus the strain-stress relations may be expressed as 

'22 = sll'z2 + s12'yy-t s13'Zz' 

where the quantities S.. characterize elastic properties of the plate material. 

'1 

4 
2.1.1 The plate under bending 

The bending problem is based on the Reissner’s plate theory (51, [SI. In 

the isotropic case the equations are given by [7]. Consider a thin elastic plate 

bounded by the planes z= * h/2. The resultant forces are shown in Fig. 1, 

where M, and Vi (i, j= z y ) are respectively the moment and transverse shear 

resultants. 

Referring to [8], the equilibrium equations may be expressed as 

vz=o, 

aMZ aMzy 

aZ ay 

-+-- 

av2 avu 

-+ -+ q = 0, a2 ay 

and, in terms of the moment and shear resultants, the stress components 

are given by 

-- , u =- u =- 

uzz- h3/12 h3/12 I ” h3/12 ’ 

Also, the following relations between some average values w, C#J~ and 4, 

and the displacements u,,, vo and wo are introduced in accordance with the 

balance of the work done by the resultant forces on the average values and by 

the corresponding stresses on the actual displacements 

5 
3 +h/2 4Z2 

w = -1 w0[ I---]dz 

2h -h/2 h2 

From (1) and the strain-displacement relations it follows that: 

awo avo ' =-+-=Sa YZ afl aZ 44 yz' 

If we now solve (5) for uij1( ilj=zlylz), substitute into (3), integrate in z 

and use (4) we obtain: 

6 

\ 
h3 adZ 

M = - (s6,)-’( - + -) ’ 

zy 12 ay a2 

aW 6vU q$=--+s - 

ay 44 5h * 

Technically, equations (2) and (6) would give the complete formulation of 

the problem, the eight equations accountling for the eight variables, M,, Ma,, 

Mzy, V,, Vu, I$,, 4u and w. Eliminating first, 4, and 4, then M,, Mu and MZu 

and assuming q=O , we obtain the system of equations governing the small 

deflection of elastic orthotropic plates as follows: 

vz,z + V&a, = O ’ 

where the coefficients Ai are constants defined in Appendix 1. The 

remaining unknowns can be obtained from 

7 
where n = S,S2, - S,,S12 . (9) 

Note that since the basic system of equation (7) is sixth order, three 

conditions must be prescribed on the boundary of the plate. 

Following 181, to simplify this bending problem a stress function F(z,y) is 

introduced as follows: 

= F,y+AI%zy + A2F,yyV ' 

then, equations (7) are identically satisfied provided the stress function 

F(z,y) satisfies the following equation: 

8 
where Bi are known constants defined in Appendix 1. 

Equation (11) is called the fundamental equation. It is essentially a special 

case of that given in [8]. 

2.1.2 In-plane stretching problem 

Begining with the usual plane stress 

uzz-uzz=u - =o, ZlI 

we define the Airy stress function Q 

assumption 

z,y) as follows: 

a1 4 

az2 ' zy ayaz g =--, 

d20 a2a 

aY2 , u =- -- 

uzz - 

Along with the stress-strain relation (l), the compatibility condition for an 

orthotropic plate may be expressed as: 

a4a a4a a4a 

aZ4 aY4 -+2c1- az2ay2 + c2 - -0, - 

The fourth order differential equation (13) is subject to two condit,ions on 

As we know, the solutions satisfying equation (13) yield the following 

characteristic equation 

m4 + 2c1 m2 + c,~=o. (13.1) 

9 
I 

The roots of (13.1) are 

m1= 41 + i4, 

"2 = 43 t- i4, 

m 2 =-m1 , m =-rn 3 4 

(13.2) 

Where ml and m2 are both real or complex conjugates. The material is 

defined as type 1 when both roots are real and as type 2 when they are 

complex conjugates. 

Equations (11) and (13) give the governing equations for an orthotropic 

plate subjected to bending and in-plane streching. 

2.2 Infinite plate containing a series of collinear cracks 

2.2.1 The orthotropic plate under bending 

Consider now an infinite plate containing a series of collinear through 

cracks which are along the y axis. This problem will be solved by using Fourier 

transforms. It will be assumed throughout this study that through a proper 

superposition, the original crack problem has been reduced to a perturbation 

problem in which self-equilibrating force and moment resultants acting on the 

crack surfaces are the only imnzer~ external loads. One may note that since the 

corresponding through crack problem is uncoupled through the governing 

equations as well as the boundary conditions, the in-plane loading and bending 

problems can be solved independently. 

For the function F(z,y) defined by (11) we introduce the following Fourier 

transformat ions: 

10 
I 

1 +a 

2r -- F(z,y) =-/ f(zla) 

Substituting (15b) into (1 1) one obtains a sixth-order ordinary linear 

differential equation in f. Looking for a solution in the form: 

!(z,a) = R(a)cm=, (16) 

the charcteristic equation of the problem is found to be 

B4m6+ ( B, - a'~~) m4 - ( a2~2 - a4~6) m2 + ( a4~3 - a6~,) = 0. (17) 

It should be emphasized here that the roots of (17) are in general complex 

and are functions of a. 

After solving (17) let the roots be ordered such that 

qmi) <OI (+1, 2, 3) mj+3=- mi. (18) 

The solution f(z,a), satisfying the regularity condition at z = 5 00, may 

then be expressed as : 

Because of the assumed symmetry with respect to yz plane in loading and 

11 
symmetry conditions: 

Vz(Z1V) = -Vz(-Z,Y), 

geometry, the transverse shear and moment resultants must satisfy the following 

It is therefore sufficient to consider one half of the plate only. Thus for 

example for z > 0 we find 

Substituting from (21) into (10) and using (8) the relevant components of 

moment and transverse shear resultants and rotation may be expressed as 

follows: 

h3 h3 h2 h2 r f-)--l-=.Lf-~ --c p. \f,j3.+-c A m3~e21 - zy\--/ - I "44 "4 J 1 

6SS6 J ' '6S66"1 loS66 

"55 3' ' 1' 

h3 h2 

12 
I 

where R,, R, and R, are unknown functions of a and are given in 

Appendix 2. 

In addition to the regularity condition at infinity, the bending problem 

must be solved under the following boundary conditions: 

13 
where (L+L’) = (-oo,+oo), L refers to a system of collinear cracks and is 

finite and p,(y) is a known function . By using the homogeneous conditions 

(26), two of the three unknown functions Rj can be eliminated. The third one is 

then obtained from the mixed boundary conditions (27). 

The problem may be reduced to an integral equation for a new unknown 

function G, defined by 

a 

--4$AY) = G,(Y), --oo < y < +0O. (28) aY 

Thus, from(27) by noting that G,(v) = 0 on y E L‘ and by using (23) and 

(25) the first equation of (27) may be expressed in the following form: 

r r+m 

The possible singularity of the kernels in (29) at y=t would be due to the 

behavior of H(a,x) as a-+*oo. For the purpose of examining the singular 

behavior of the kernels in (29) and for extracting the singular parts, the 

asymptotic behavior of m. as 101 -00 is needed. Thus, from (17) it can be 

shown that for large values of 1.1 we have 

J’ 

14 
3' m =-m m3= (a2 - ql4' 6 

( 30) 

where ai and bi are material constants. 

Using now the relations (30) and separating the asymptotic value of H for 

large IaI, the kernel in (29) may be expressed as 

where H" is the asymptotic value of H(a,z) for /a/ -+ 00. The first term 

on the right-hand side of (31) gives a Canchy-type kernel (t-y)-' and the 

second integral is uniformly converged for all t and y in which the limit z=o 

can, therefore, be put under the integral sign. After some rather complicated 

and lengthy manipulations (mostly due to the large number of elastic constants) 

(29) may be shown to be a singular integral equation with a simple Chachytype singularity of the following form : 

where pl is a material constant and 

15 
K(y,t) = H(a) - P(a)] e'a'('-Y)da. (33) -W 

If ci < 0 < dil (;=I, . . . ,n) defines the cracks along the y axis, from 

(27) and (28) it follows that (34) must be solved under the following singlevaluedness conditions: 

For example, if the plate contains a single crack along (-ala) subjected to 

uniform bending moment 

P,M = -M"1 (35) 

it is seen that ,!,=(-ala) and the solution of the integral equation may be 

expressed as [lo] 

where the unknown function g,(t) is bounded in It/ 5 a. 

Defining now 

16 
SM" 

Ub = - 

h2 ' (37) 

equation (32) may be expressed as 

J_:" 

dl - 72 

1 -+ k( 6,7)] d7= -1, 7- 8 

subjected to 

Lg dr= 0. (39) 

In the symmetric bending problem under consideration, the stress intensity 

factor at a crack tip y=d, is defined as follows: 

k,(z) = lim [ 2 ( y - d,) ] 1/2 uz (O,y,z). 

p+ d. 

Refering to for the procedure, in this single crack case the stress 

intensity factor ratio defined 

2 

kl(2) =-k u &, h/2 bb 

is found to be 

kbb = -xg(I) . 

17 
We must emphasize again that the major difficulty in this problem is that 

the functions m{a) are not known explicitly in terms of (a). Thus we must 

make use of a numerical technique to obtain H(a) and then, k( vlt). 

J 

2.2.2 The Orthopropic Plate under In-Plane Loading 

Similarly for the in-plane loading problem we let 

Substituting now from (43) into (13) and considering the symmetry 

conditions, we obtain: 

for materials of type 1 

421.) = n,(4 exP(q3 lot z 1 + n,(Q) exp(q4 IQI 2 11 (2’ 0) (44.1) 

where 

rn, = q3 = - ( C, + Jc12 - c,’ I/’, 

m2 = q4 = - ( C, - Jc12 - c22 ) 112. (44.2) 

are reai roots of the characteristic equation ji3.i) and 2, and C, are 

given by (14). 

for materials of type 2 

d(.,.) = [ nl(a)cos ( 92Q= ) + R,(.)sin (y. 1 1 exP(ql IQI z 1 > 

(2 ’ 0) (45.1) 

where 

18 
(45.2) 

are complex conjugate roots of the characteristic equation (13.1). 

The membrane problem must be solved under the following boundary 

condition: 

Nzy(0’ar) = 01 (46) 

(47) 

One of the two unknown functions R, and R2 may be eliminated by using 

(46) and the remaining one may then be obtained from the mixed boundary 

conditions (47). Defining a new unknown function G2 by 

i using the following expressions obtained from (13), (43) and (44) or (45), 

r-- 1u1 -.-.r..,:, ,I,o”L*Iu!IJ P cf typo 1 

19 
(49.1) 

+ n2(a) exp(q4)al z) ] e-iayda 

€or materials of type 2 

+ n2(a) I q2 a cos (q2 a z ) + qllal sin (q2 a z ) ] } exp(ql la/ z ) e-ia%ia 

(49.2) 

1 +O0 

u =-J (-a2)[nl(a)cos(q20z) 

= 2a -oo 

and noting that 

one may obtain the following integral equation to determine G, 

20 
‘I 

where for materials of type 1 

for materials of type 2 

(52.1) 

Equation (51) must be solved under the following single-valuedness 

conditions, 

2.3 The line-spring model for surface crack 

The most general description of the line-spring model introduced in (11 has 

been given in 1111. In a plate containing a part-through crack and subjected to 

membrane and bending loads, the net ligament (the uncracked portion) around 

the crack would generally have a constraining effect on the crack surface 

displacement. By approximately representing the net-ligament stress as a 

membrane load N and a bending moment M and the crack surface displacement 

due to N and M as an opening 6 and a rotation 0, the three-dimensional crack 

problem could be reduced to a two-dimensional coupled bending-membrane plate 

21 
problem. Furthermore, assuming that the relationship between (N,M) and (a$) 

may be found through the plane strain results obtained from the solution of an 

edge-notched strip, the pair of functions (a$) and (M,N), then, are determined 

from the corresponding mixed boundary value problem for a plate having a 

through crack in which N and M are treated as unknown crack surface loads. 

Let the stress intensity factor for the plane strain problem in Fig. (2.b) be 

given as 

where functious gb(s) and g,(1) are called the shape functions for tension 

and bending, respectively and given as: 

i=l 

(56.a) 

(56.6) 

The coefficients AT, and AB, can be found by applying a suitable curve 

fitting to the results obtained from [I21 which are valid for 

0 < L,/h < 0.8. 

Referring to [13], the strain energy release rate G, in an orthotropic 

medium may be obtained as: 

22 
1 1 1 

11 

(57) 

where K, is the stress intensity factor and the elastic constants bij are 

givens by (see equation(1)) 

'33'22-'23 

b= 

s22 

11 I 

Sl 1 s22 - '1 22 

b= 

s22 

22 1 

'1 3 '22- '1 2 '23 

s22 

'66 = 1 

Also, the energy available for fracture can be expressed as: 

a 1 a6 ae 

Gl=-(U-V)=-(N-+M~). f3L 2 f3L aL 

Combining now (57) and (59) and using (54) we obtain 

From (60) it can be shown that 

(59) 

23 
t 

1 

6 = 2h - ( atbm + atfu ) , p3 

where Q.. are the compliance coefficients and are given by 11 

From (28) and (48) it may be seen that 

6 = 2u( O,y) = 2 G2( t )dt . l;: 

Solving (61) for u and m and substituting from (63), we obtain 

24 
2 A=Q Q -Q~~ bb ti 

One should keep in mind that a,,, consequently 7,j are fuctions of L(y)/y 

and that L(y) describes the surface crack profile. 

As an example let us now consider a plate containing a single surface 

crack, as shown in Fig. 2. By using this model, the crack may be assumed to 

be a through crack of length 2a and the constraint caused by the net ligament 

may be accounted for by applying the membrane and bending resultants N(y) 

and M(y) on the crack surfaces. One may note that these net ligament stresses 

tend to prevent the crack face from opening and rotating. The mixed boundary 

condition corresponding to (27) and (47) may be expressed as : 

M,, = Gl(Y) + M(Y) 1 -a<y<a 

(66) 

4, (0,Y) = 0 ’ IYI ’ 

u(0,y) = 0. IYI > a 

The integral equstions, corresponding to (32) and (51) then become : 

25 
where 

(68.a) 

(68.6) 

(68.c) 

(68.d) 

As seen from (68) in the surface crack problem the integral equation for 

bending and tension are coupled. 

2.4 Two special cases of surface cracks 

As mentioned earlier, integral equations for the general case given in (32) 

and (51) are valid for any number of collinear cracks. But in this study, for the 

purpme of simp!ificat,inn, we only consider two special cases: a single symmetric 

crack and two arbitrarily arranged collinear cracks. 

In the first case, referring to Fig. 2, assuming the plate subjected to the 

uniform bending MaO and tension N" and introducing the following change in 

variables 

26 
Y 

a 

a=- -1 < (7,1) < 1 , 

from(68), we obtain: 

(70.a) 

(70.b) 

It is seen that (70) has a simple Cauchy kernel and hence the solution of 

the integral equation is of the following form : 

hj(4 

V'l - 2 

9j(7) = = , (i= 1,2) , -1 < r < 1. 

Equations (70) may be solved numerically by using the quadrature 

formulas given , for example , in 191, under the additional conditions: 

g;(r)dr = 0, (i= 1,2). J_: 

27 
After solving (70) for g1 and g,, the stress and moment resultants M(y), 

N(y) may be obtained from (64), and the stress intensity factor K(y) along the 

crack front is then determined from (54). 

The configuration and parameters of the second special case are shown in 

Fig. 3. In this case the unknown functions Gi should be defined separately for 

cracks (A,B) and (C,D) as: 

G,, : derivative of the crack surface rotation for the crack (A,B); 

G,, : derivative of the crack opening displacement for the crack (A,B); 

G,, : derivative of the crack suface rotation for the crack (C,D); 

G2, : derivative of the crack opening displacement for the crack (C,D); 

also let N(x) and M(x) for these two cracks be defined separately as follows : 

N,(x) : N(x) for the crack (A,B); 

N2(x) : N(x) for the crack (C,D); 

M,(x) : M(x) for the crack (A,B); 

M,(x) : M(x) for the crack (C,D). 

For crack (A,B) the system equation can then be written as: 

A <y, <B 

28 
similarly for crack (C,D) we have 

where 

Again, we have to convert the limits of integrals to (-1,l) for mumerical 

sc!utIm by introducing the fo!!nwing parameters: 

29 
B-A B+A 

t =- rl+-- - ~7~ + (a + d/2) , l2 2 

B-A B+A 

S1 +-- -Q61+(~+d/2), 2 Y, = - 2 

D-C D+C 

t -- r2+-- -cr2+(c+d/2), 2- 2 2 

t 

D-C D+C "+-- - + ( c + d/2 ) , 2 Y2 = - 2 

Substituting from (75) into (73) and (74) we find 

30 
-1< s1 < 1 

-1< Sl < 1 

where the kernels F,, F,, F, and F, are bounded in the close interval 

31 
(i=1,2). 

This problem has four integral equations (77) -(80) with four unknown 

functions g,,, gI2’ g2, and gZ2 which may be solved by using Gass-Chebyshev 

closed-type quadrature formula under the following single-valuedness conditions : 

1; pij ( t )dt = 0 ’ ( i, i= 1,2) . 

After solving for the unknown functions, the same procedure for single 

crack may be used for each crack separately in order to find stress intensity 

factors. It may be observed that if we let Mi = 0 , Nj = 0 or 7..=0, the 

integral equations (77)-(80) reduce to the system for two collinear through 

cracks in which the pair of equations corresponding to bending and tension 

would be uncoupled. One may also observe that as the distance between the 

F vanish and the 

integral equations corresponding to cracks (A,B) and (C,D) would again be 

uncoupled. For through cracks in the latter case one would obtain four 

uncoupled equations. 

‘3 

two crack tends to infinity, the Fredholm kernels F,, - - - ’4 

2.5 Some remarks on the llbrniiiiation of the orihoiropic plate 

problem 

In the previous section the general problem for an orthotropic plate is 

formulated. The formulation is given for an orthotropic material which is 

defined by Hook’s law as expressed in (1). Since this is the first attempt to 

solve this kind problem, it is important to make the following remarks: 

32 
I 

1. Unlike the classical orthotropic plate theory, here six rather than four 

material constants have been used to formulate the plate bending problem: four 

to describe the in-plane deformations names Ex, Ey, vxy, and Gxy, (in terms of 

engineering material constants) and two to describe the out-of-plane 

deformations, namely Gxz and Gys. In the classical bending theory GXL and Gyr 

are assumed to be infinite. 

2. Making use of the line-spring model in addition to the plate problem, 

an approximate solution is given for the three dimensional part-through crack 

problem. To do this the remaining three material coustants Ez, vzx, vzz 

(referring to Fig. 2) have also been introduced to the formulation. Thus, all 

nine material constants of orthotropy must be specified in order to solve the 

surface crack problem. 

3. As we mentioned earlier, if the plate material is isotropic, instead of 

one sixth order fundamental equation given by (ll), the governing equation may 

be exprcssed in terms of one fourth order and one second order differential 

equation (see, for example (21). Thus, it is analytically impossible to arrive at 

the same isotropic plate formulation as given in [2] by taking the limiting case 

of the orthotropic plate formulation given in this study. However, good 

agreement is obtained with the result given in [2] by using elastic constants in 

the numerical solution of the orthotropic plate problem which are nearly the 

same as the isotropic constants. It may also be noted that if the isotropic 

material constants are substituted into the characteristic equation (17), as 

expected the roots turn out to be the same as those found in the isotropic case. 

33 
Chapter 3 

Results and discussion 

The main interest in this study is in evaluating the stress intensity factor 

in an orthotropic plate. The similar study for the isotropic plate has been 

considered before in [2] and [IO]. However, some basic and additional results 

for this case have been studied for the purpose of comparison and extension. 

The elastic constants of the materials used in the numerical examples are 

given in Table 1. Material 1 is basically isotropic, and 2 and 3 are laminated 

composite material. Except for a 90 degree rotation of the axes of orthotropy, 

Material 2 and 3 are identical. Most of the results obtained are for loading 

conditions 

N(Y) = N"0, M(Y) = 0 9 (tension) 

M(y) = M", N(y) = 0. (bending) 

The first case that is studied is the plate containing a through crack and 

two collinear through cracks. Because of the nature of the plate theory used in 

analysis, the stress intensity factor is a linear function of the thickness 

coordinate z [see (41)]. 

Table 8 and Fig. 4 show the effect of the thickness ratio on the stress 

int,ensity factor for a single through crack. The result for the isotropic case 

(v=0.3) are the same as that in [lo]. It is clearly seen that the orthotropic 

materials have relatively higher stress intensity factors . This is due to a higher 

GZy in these orthotropic cases. 

Some numerical results obtained for two arbitrarily arranged collinear 

34 
through cracks in a plate under uniform bending are shown in Tables 9-14 and 

Figures 5-6. Tables 9-11 show the interaction between two cracks with the 

length of one crack varying for Material 1, 2 and 3 and Tables 12-14 show 

similar results with the plate thickness varying. Tables 15-16 show the results 

for a plate under uniform tension. As expected for the distance between the two 

cracks d '00 the single crack results are recovered, and the stress intensity 

factor at the inner tip is always greater than at the outer tip. Figures 5 and 6 

clearly show the effect of bending due to the nonsymmetric orientation of the 

two cracks in the y direction. It is again noted that for the in-plane stretching 

problem the material orthotropy has no effect on the stress intensity factor for 

an infinite orthotropic plate containing a single crack or series of collinear 

cracks. 

After through cracks, the part-through crack problem is considered. 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 the 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 , referring to Fig. 2(a), is defined by 

L(Y) = L& - (y/aI2 1 -a < y< a 1 

Lo being the total crack depth at the midsection ( y = 0 ). 

Tables 17-19 show the stress intensity factor at y= 0 in an infinite plate 

containing a single crack under uniform tension and pure bending for Materials 

1, 2 and 3 respectively. In these cases, the stress intensity factors are 

normalized with respect to K" which is the corresponding value for an edge 

crack strip under plane strain condition with the same L,/h ratio [Fig. 2(b)]. 

35 
Note that the limiting values of these stress intensity factors are 

a 

K- 0, for -- h 0, 

K- Km, jor --+ h 00. 

a 

It should be noted in this case that because of crack closure on the 

compression side, taken separately the bending results are meaningless. 

However, they may be used by superposition with tension results which are 

sufficiently large so that the stress intensity factor on both sides of the crack 

are positive. The coefficients of the shape functions gb and gt for these three 

materials are listed in Tables 3, 5 and 7 which is based on the stress intensity 

factors shown in Tables 2, 4 and 6. 

Figures 7 and 8 clearly show the comparison of different material on the 

stress intensity factor for L,/h varying from 0.2 -0.8. As expected under 

bending the crack faces would be closed on the compression side; therefore, for 

deep cracks the stress intensity factor in an isotropic medium becomes negative. 

From Tables 18 and 19 and Figures 8 and 9 we have found that the stress 

intensity factor obtains the negative value as the L,Jh larger than 0.5. 

Some results regarding the distribution of the normalized stress intensity 

factor at the crack front for a single semi-elliptic surface crack are given in 

Table 20. 

The results for two collinear semi-elliptic surface cracks are shown in 

Tables 21-26. Here the crack profile is defined by 

36 
L( y) = L,J1- y2 , 

Results for the interaction of these two collinear, semi-elliptic surface 

cracks with the distance between them varying from d/a=0.1 - 00 and the 

length of one crack varying from c/a=0.1 - 1. under tension or bending have 

been tabulated in Tables 21-23 for material 1, 2 and 3 respectively. The effect 

of L,/h on the normalized stress intensity factor for the case of these two 

collinear cracks has also been considered. Tables 24-26 show this effect for the 

case of two identical collinear cracks. It may be observed again from all of 

these tables that the material orthotropy does have significant effect on the 

fracture parameter for the surface crack. 

Finally, an additional example has been given for a crystalline material 

TOPAZ [ Sio4AL2 (F,OH), ] which has an orthohomibiz system behavior. The 

material constants of TOPAZ, given by 1141, are as follows: 

(unit cm2/dyne ) 

Material 4 

s,,=4.43 

S4,=9.25 

S, 2=- 1.38 

s2,=3.53 S,,=3.84 

S5,=7.52 S,, =7.63 

S 3=-0. 86 S2,=-0.66 

37 
Material 5 

s,,=3.53 s,,=4.43 S3,=3.84 

S4,=7.52 S,,=9.25 S6,=7.63 

S, 2=- 1.38 S,,=-0.66 S,,=-0.86 

Note that the only difference between Material 4 and Material 5 is a 90 

degree rotation of orthotropy. Table 27 shows the effect of the thickness ratio 

a/h on the stress intensity factor in a single through cracked plate under 

uniform bending for Material 4 and Material 5. and Table 28 shows the effect 

of L,/h on the normalized stress intensity factor at the deepest penetration 

point of a semi-elliptic surface crack under uniform tension and bending 

moment. It may be seen from these tables that there is no important difference 

between these two kinds of orthotropy orientations, as this material does not 

have strong orthotropic material properties. 

I 

38 
Tables and Figures 

Table 1: Elastic constants of the materials used in the examples 

Mat.1 Mat.2 Mat.3 

Ex psi 2.2447*106 5.86.1 O6 22.2.106 

GPa 15.477 40.405 153.069 

EY psi 2.26-1 O6 22.2- 1 o6 5. 86-106 

GPa 15.583 153.069 40.405 

EL psi 2. 26.106 3.3-106 3.3-106 

G Pa 15.583 22.754 22.754 

Gxy Psi 0.866~10~ 4.25- 1 O6 4.25*106 

G Pa 5.971 29.304 29.304 

Gyc Psi 0.866.1 O6 0.592-106 0.225.106 

GPa 5.971 4.082 1.551 

0.866- lo6 0.'L25-1U6 0.592.iijG GXL Psi 

GPa 5.971 1.551 4.082 

v 

v 

vXC 

XY 

YE 

0.3 

0.3 

0.3 

0.484 1.834 

0.195 0.261 

0.261 0.195 

39 
Table 2: The coefficients of AT, and AB, for the shape 

functions g, and g, ( Material 1) 

1 AT, AB, 

1 1.122 1.120 

2 6.520 -1.887 

3 -12.388 18.014 

5 -188.608 24 1.912 

7 -32.052 168.01 1 

4 89.055 - 8 7.3 8 5 

6 207.387 -319.940 

Table 3: Stress intensity factors in an edge crack under 

tension (N) or bending (M)7 

(u=;, m=- 

(Material 1) 

6M 1 N 

h2 

L 

h 

- KM 

mG 

0.001 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

1.122 

1.189 

1.367 

1.660 

2.111 

2.825 

4.033 

6.355 

11.955 

1.120 

1.047 

1.055 

1.124 

1 .%I 

1.498 

1.915 

2.728 

4.691 

40 
Table 4: The coefficients of AT, and AB, for the shape 

functions g, and g, ( Material 2) 

1 AT, AB, 

1.047 1.043 

-27.969 17.276 

-439.451 232.556 

-222.80 158.307 

7.639 -1.610 

175.360 -84.989 

557.540 -304.196 

Table 5: Stress intensity factors in an edge crack under 

tension (N) or bending (M), 

(u=,, m=- ) 

(Material 2) 

N 6M 

h2 

L 

h 

- 

KM 

0.001 

0.1 

0.2 

0.3 

9.4 

0.5 

0.6 

0.7 

0.8 

1.042 

1.129 

1.318 

1.607 

2.042 

2.721 

3.860 

6.038 

11.277 

1.041 

0.992 

1.013 

1.083 

1.21 1 

1.430 

1.814 

2.563 

4.372 

41 
Table 6: The coefficients of AT, and AB, for the shape 

functions g, and g, ( Material 3) 

I 

1 AT, AB, 

1 1.055 1.051 

3 -25.426 17.517 

5 -396.808 234.182 

7 -191.359 159.403 

2 7.461 -1.664 

4 160.659 -85.71 1 

6 498.577 -306.252 

Table 7: Stress intensity factors in an edge crack under 

tension (N) or bending (M), (Material 3) 

L 

h 

- 

N 6M 

h2 

(u=;, m=- ) 

0.001 1.050 1.049 

0.1 1.134 0.996 

0.2 1.321 1.016 

0.3 1.611 1.086 

0.4 L.U*O I.LIJ 

0.5 2.730 1.436 

0.6 3.876 1.823 

0.7 6.068 2.577 

0.8 11.350 4.402 

n nin 1 OlC 

42 
Table 8: The effect of the thickness ratio on the stress intensity 

factor in a plate under uniform bending, containing 

a single crack, ub= 6M,/h2 

a 

h - Mat.1 Mat.2 Mat.3 

0.05 

0.1 

0.25 

0.5 

1 

2 

4 

6 

10 

- 

kl 

Ob< 

0.9885 

0.9676 

0.8992 

0.8195 

0.7477 

0.7003 

0.6701 

0.6446 

0.6481 

0.9992 

0.9974 

0.9886 

0.9689 

0.9276 

0.8664 

0.8086 

0.7776 

0.7166 

0.9987 

0.9957 

0.9823 

0.9548 

0.9050 

0.8434 

0.7942 

0.7657 

0.7044 

43 
I 

Table 9: Stress intensity factors in a plate containing two collinear cracks 

~ subjected to uniform bending moment M, ( Material 1) 

B-A ( a=- 

2' 

D-C 

c=- 

2' d=C-B, "1 ) h 

- 4 0.1 0.25 0.5 1 2 00 

d 

a 

E 

- 

a 

1 0.8799 0.8551 0.8313 0.8045 0.7798 0.7477 

0.5 0.8071 0.7938 0.7821 0.7698 0.7593 0.7477 

0.25 0.7711 0.7647 0.7598 0.7551 0.7513 0.7477 

- 

klA 

O< 

0.1 0.7532 0.7512 0.7500 0.7490 0.7482 0.7477 

1 1.294 1.076 0.9599 0.8697 0.8049 0.7477 

klB 

- & 0.5 1.063 0.9143 0.8458 0.7995 0.7698 0.7477 

uva 

0.25 0.9161 0.8220 0.7863 0.7663 0.7550 0.7477 

0.1 0.8088 0.7678 0.7563 0.7514 0.7489 0.7477 

1 1.294 1.076 0.9599 0.8697 0.8049 0.7477 

klC 

- k 0.5 1.012 0.8405 0.7498 0.6786 0.6261 0.5794 

uva 

0.25 0.7990 0.6595 0.5867 0.5297 0.4872 0.4496 

0.1 0.5647 0.4577 0.4037 0.3627 0.3325 0.3060 

1 0.8799 0.8551 0.8313 0.8045 0.7798 0.7477 

0.7395 0.7071 0.6771 0.6434 0.6132 0.5794 kl D 

- 

& 0.5 ova 

0.25 0.6275 0.5867 0.5507 0.5135 0.4816 0.4496 

0.1 0.4817 0.4293 0.3917 0.3577 0.3308 0.3060 

44 
t 

I 

Table 10: Stress intensity factors in a plate containing two collinear cracks 

' subjected to uniform bending moment M, ( Material 2) 

B-A D-C d=C-B, "1 ) 2' ==- 2' h ( a=- 

d 

--0.1 0.25 0.5 1 2 00 

1 1.051 1.023 1.001 0.9782 0.9590 0.9276 

klA 

- 0.5 0.9814 0.9658 0.9546 0.9451 0.9373 0.9276 

0.25 0.9488 0.941 1 0.9363 0.9328 0.9305 0.9276 

U4 

0.1 0.9329 0.9305 0.9293 0.9285 0.9280 0.9276 

1 1.591 1.274 1.122 1.029 0.9773 0.9276 

klB 

- 0.5 1.320 1.106 1.015 0.9665 0.9441 0.9276 

0.25 1.137 1.008 0.9614 0.9405 0.9325 0.9276 

U& 

0.1 1.003 0.9500 0.9353 0.9301 0.9284 0.9276 

1 1.591 1.274 1.122 1.029 0.9773 0.9276 

klC 

UC 

- 0.5 1.221 0.9665 0.8424 0.7658 0.7240 0.6851 

0.25 0.9261 0.7202 0.6188 0.5567 0.5236 0.4943 

0.1 0.6327 0.4774 0.4023 0.3577 0.3348 0.3154 

1 1.051 1.023 1.001 0.9782 0.9590 0.9276 

klD 

- 0.5 0.8441 0.8019 0.7687 0.7386 0.7151 0.6851 UG 

0.25 0.6817 0.6228 0.5790 0.5436 0.5201 0.4943 

0.1 0.5143 0.4375 0.3880 0.3535 0.3335 0.3154 

45 
.. 

Table 11: Stress intensity factors in a plate containing two collinear cracks 

sobjected to uniform bending moment M,, ( Material 3) 

B-A 

2' ( a=- D-C 

c=- 

2' d=C-B, "1 ) h 

- -+ 0.1 0.25 0.5 1 2 00 

d 

C 

- 

a 

1 1.031 1.004 0.9821 0.9582 0.9375 0.9050 

0.5 0.9595 0.9442 0.9332 0.9238 0.9152 0.9050 

0.25 0.9263 0.9188 0.9141 0.9106 0.9083 0.9050 

kl A 

UG 

- 

0.1 0.9194 0.9080 0.9068 0.9060 0.9059 0.9050 

1 1.547 1.246 1.101 1.011 0.9581 0.9050 

kl B - L 0.5 1.287 1.080 0.9917 0.9452 0.9228 0.9050 UG 

0.25 1.111 0.9841 0.9385 0.9183 0.9103 0.9050 

0.1 0.9103 0.9271 0.9127 0.9076 0.9054 0.9050 

1 1.547 1.246 1.101 1.011 0.9581 0.9050 

1.206 0.9550 0.8301 0.7507 0.7067 0.6752 klC 

- 0.5 I0.1 0.6308 0.4775 0.4019 0.3555 0.3311 0.3149 

1 1.031 1.004 0.9821 0.9582 0.9375 0.9050 

0.8289 0.7869 0.7529 0.7216 0.6981 0.6752 klD 

- 0.5 QG 

0.25 0.6777 0.6187 0.5736 0.5364 0.51 16 0.4912 

0.1 0.5140 0.4375 0.3872 0.3510 0.3300 0.3149 

46 
Table 12: The effect of the thickness ratio on the stress intensity factors 

uhder uniform bending moment M, (Material 1) 

B-A D-C ( a=2’ c=- 2’ d=C-B, :=0.5 ) a 

- 4 0.1 0.25 0.5 1 2 00 d 

a 

a 

h 

- 

0.5 0.8715 0.8596 0.8502 0.8408 0.8320 0.8195 

1 0.8071 0.7938 0.7821 0.7698 0.7593 0.7477 

2 0.7625 0.7463 0.7322 0.7191 0.7094 0.7003 

- 

klA 

UC 

- 

klB 

c 

0.5 1.148 0.9757 0.9031 0.8631 0.8409 0.8195 

1 1.063 0.9143 0.8458 0.7995 0.7698 0.7477 

ut/ a 

2 1.039 0.8898 0.8071 0.7493 0.7184 0.7003 

0.5 1.082 0.8825 0.7872 0.7256 0.6841 0.6358 

1 1.012 0.8405 0.7498 0.6786 0.6261 0.5794 - 

klC 

L 

UG 

2 0.9896 0.8147 0.7061 0.6204 0.5675 0.5287 

0.5 0.7831 0.7522 0.7260 0.6989 0.6732 0.6358 

0.7395 0.7071 0.6771 0.6434 0.6132 0.5794 klD 

- 1 /- 

uva 

2 0.6933 0.6549 0.6192 0.5837 0.5561 0.5287 

47 
Table 13: The effect of the thickness ratio on the stress intensity factors 

under uniform bending moment Mo ( Material 2) 

B-A D-C 

2’ ( a=- 2, c=- d=C-B, :=0.5 ) 

- + 0.1 0.25 0.5 1 2 00 

d 

a 

h 

- 

0.5 1.027 1.009 0.9956 0.9849 0.9772 0.9689 

1 0.9814 0.9658 0.9546 0.9451 0.9373 0.9276 

2 0.9226 0.9087 0.8958 0.8890 0.8865 0.8664 

- 

klA 

04 

0.5 1.390 1.160 1.061 1.008 0.9836 0.9689 

1 1.320 1.106 1.015 0.9665 0.9441 0.9276 

2 1.227 1.036 0.9550 0.9100 0.8771 0.8664 

- 

klB 

U4 

0.5 1.279 1.002 0.8652 0.7796 0.7336 0.6990 

1 1.221 0.9665 0.8424 0.7658 0.7240 0.6851 - 

klC 

2 1.148 0.9248 0.8164 0.7462 0.7024 0.6559 

0.5 0.8692 0.8214 0.7834 0.7495 0.7250 0.6990 

1 0.8441 0.8019 0.7687 0.7386 0.7151 0.6851 - 

klD 

,- 

UVIl 

2 0.8144 0.7792 0.7463 0.7204 0.6905 0.6559 

48 
Table 14: The effect of the thickness ratio on the stress intensity factors 

under uniform bending moment M, ( Material 3) 

B-A D-C 

2’ ( a=- 2’ c=- d=C-B, :=0.5 ) 

- -.+ 0.1 0.25 0.5 1 2 00 d 

a 

a 

h 

- 

0.5 1.011 0.9942 0.9817 0.9713 0.9701 0.9548 

1 0.9595 0.9442 0.9332 0.9238 0.9152 0.9050 

2 0.9006 0.8867 0.8720 0.8671 0.8530 0.8433 

- 

klA 

UG 

0.5 1.371 1.143 1.046 0.9937 0.9638 0.9548 

1 1.287 1.080 0.9917 0.9452 0.9228 0.9050 - 

klB 

,- 

uda 

2 1.188 1.009 0.9323 0.8880 0.8638 0.8433 

0.5 1.274 1.000 0.8641 0.7773 0.7293 0.6946 

1 1.206 0.9550 0.8301 0.7507 0.7067 0.6752 - 

klC 

fuva 

2 1.106 0.8840 0.7743 0.7049 0.6668 0.6400 

0.5 0.8671 0.8193 0.7809 0.7460 0.7200 0.6946 

1 0.8289 0.7869 0.7529 0.7216 0.6981 0.6752 - 

klD 

Iuq a 

2 0.7688 0.7341 0.7056 0.6791 0.6592 0.6400 

49 
Table 15: Stress intensity factors in a plate containing two collinear cracks 

subjected to uniform tension No 

E-A D-C d=C-B, "1 ) 2' h ( a=- 

2 ' c=- 

- -+ 0.1 0.25 0.5 1 2 00 d 

a 

1 1.151 1.112 1.0811 1.052 1.028 1.000 

0.5 1.066 1.045 1.030 1.018 1.009 1 .ooo 

0.25 1.026 1.016 1.010 1.005 1.003 1 .ooo 

- 

klA 

U& 

0.1 1.007 1.004 1.002 1.001 1 .ooo 1 .ooo 

klB 

1 1.795 1.414 1.229 1.113 1.048 1.000 

0.5 1.449 1.206 1.100 1.043 1.016 1 .ooo U< 

0.25 1.234 1.090 1.038 1.014 1.005 1 .ooo 

0.1 1.083 1.025 1.009 1.003 1.001 1 .ooo 

1 1.795 1.414 1.229 1.113 1.048 1 .ooo 

1.035 0.8889 0.7962 0.7446 0.7071 klC 

UG 

- 0.5 1.329 

0.25 nc9915 0.7595 0.5955 0.5683 0.5280 0.5000 

0.1 0.6732 0.5006 0.4151 0.3624 0.3349 0.3162 

1 1.151 1.112 1.0811 1.052 1.028 1.000 

0.5 0.8956 0.8429 0.8007 0.7626 0.7347 0.7071 

0.25 0.7170 0.6480 0.6425 0.5524 0.5235 0.5000 

- 

klD 

Udi 

0.1 0.5422 0.4557 0.3985 0.3573 0.3336 0.3162 

50 
I 

Table 16: The effect of the thickness ratio on the stress intensity factors 

under uniform tension No 

B-A D-C 

2’ ( a=- d=C-B, :=0.5 ) 2’ c=- 

d 

- -+ 0.1 a 

a 

h 

- 

0.25 0.5 1 2 00 

0.5 1.066 1.045 1.030 1.018 1.009 1 .ooo 

1 1.066 1.045 1.030 1.018 1.009 1 .ooo 

2 1.066 1.045 1.030 1.018 1.009 1 .ooo 

- 

klA 

U4 

0.5 1.449 1.206 1.100 1.043 1.016 1 .ooo 

- 1 1.449 1.206 1.100 1.043 1.016 1 .ooo UG 

2 1.449 1.206 1.100 1.043 1.016 1 .ooo 

klB 

0.5 1.329 1.035 0.8890 0.7962 0.7446 0.7071 

1 1.329 1.035 0.8890 0.7962 0.7446 0.7071 

2 1.329 1.035 0.8890 0.7962 0.7446 0.7071 

klC 

U& 

- 

‘lD 

0.5 0.896 0.843 0.801 0.763 0.735 0.7071 

1 0.896 0.843 0.801 0.763 0.735 0.7071 

UG 

2 0.896 0.843 0.801 0.763 0.735 0.7071 

51 
Table 17: Stress intensity factors at the maximum penetration point of 

a semi-elliptic surface crack in an infinite plate under uniform 

tension or pure bending (Material 1) 

0.1 0.378 0.060 -0.039 -0.0296 

0.25 0.585 0.184 -0.003 -0.0345 

0.5 0.715 0.309 0.053 -0.0287 

1 0.811 0.446 0.136 -0.0119 

- 

Klb 

KW 

2 0.882 0.586 0.242 0.0179 

0.1 0.423 0.176 0.079 0.021 

0.25 0.613 0.284 0.124 0.034 

0.5 0.733 0.392 0.176 0.050 

1 0.823 0.510 0.247 0.073 

2 0.889 0.631 0.339 0.104 

- 

Klt 

KW 

52 

L 
Table 18: Stress intensity factors at the maximum penetration point of 

a semi-elliptic surface crack in an infinite plate under uniform 

tension or pure bending (Material 2) 

0.25 0.518 0.130 -0.025 -0.036 

0.5 0.675 0.255 0.019 -0.036 

1 0.798 0.409 0.097 -0.026 

2 0.873 0.559 0.205 -0.0005 - 

Klb 

Km 

4 0.924 0.693 0.339 0.045 

6 0.344 0.759 0.425 0.082 

10 0.962 0.829 0.536 0.142 

0.25 0.552 0.237 0.102 0.027 

0.5 0.697 0.345 0.147 0.039 

1 0.810 0.477 0.215 0.058 

- 2 0.881 0.608 0.307 0.086 

KW 

4 0.928 0.726 0.422 0.129 

6 0.947 0.784 0.496 0.163 

10 0.965 0.846 0.591 0.217 

53 
Table 19: Stress intensity factors at the maximum penetration point of 

a semi-elliptic surface crack in an infinite plate under uniform 

tension or pure bending (Material 3) 

0.25 0.629 0.212 0.0024 -0.037 

0.5 0.765 0.361 0.069 -0.031 

1 0.861 0.521 0.172 -0.103 

2 0.913 0.660 0.299 0.028 

4 0.948 0.775 0.446 0.090 

6 0.963 0.829 0.534 0.139 

10 0.975 0.882 0.640 0.215 

Klb 

~ 

KW 

0.25 0.654 0.308 0.131 0.035 

0.5 0.780 0.435 0.191 0.051 

1 0.869 0.574 0.279 0.076 

- 2 0.918 0.696 0.387 0.1 14 

KW 

4 0.952 0.798 0.5 13 0.170 

6 0.965 0.846 0.589 0.214 

10 0.977 0.894 0.681 0.281 

54 
Table 20: Normalized stress intensity factors at the crack front 

for a semi-elliptic surface crack with a/h = 1 

1 

Mat.1 Mat.2 Mat.3 II 

a 

- 

0.924 0.509 0.498 0.544 

- 

Klb 

KLO 

- = 0.4 h 

0.707 0.481 0.453 0.538 

0.383 0.450 0.416 0.522 

0.000 0.446 0.409 0.521 

0.924 0.396 0.391 0.419 

- 0.707 0.453 0.433 0.498 _. 

K0.383 0.491 0.461 0.549 

0.000 0.510 0.477 0.574 

LO 

- = 0.6 h 

0.924 

0.707 

K0.383 

0.000 

0.924 

- 0.707 

Km 

0.383 

0.275 

0.21 1 

0.154 

0.136 

0.210 

0.224 

0.239 

0.254 

0.183 

0.121 

0.097 

0.200 

0.203 

0.2 12 

0.321 

0.257 

0.195 

0.172 

0.243 

0.258 

0.273 

0.000 0.247 0.215 0.279 

55 
Table 21: The interaction between two cracks on the normalized stress 

intensity factors at the deepst penetration point of a semielliptic surface crack under uniform bending or tension 

(Material 1) 

B-A D-C a a =-, 2 c =- ,d=C-B, -=I h 

- + 0.1 0.25 0.5 1 2 00 

d 

a 

C 

- 

a 

1 0.313 0.306 0.299 0.290 0.283 0.272 

0.5 0.292 0.287 0.282 0.278 0.274 0.272 

0.25 0.280 0.275 0.275 0.273 0.272 0.272 

(B) 

- 

KOab 

KW 

0.1 0.273 0.273 0.272 0.272 0.272 0.272 

1 0.313 0.301 0.299 0.290 0.283 0.272 

- 0.5 0.197 0.188 0.179 0.171 0.164 0.160 

K= 

0.25 0.101 0.091 0.083 0.076 0.072 0.069 

KOP 

0.1 0.012 0.0045 -0.0004 -0.0038 -0.0057 -0.0058 

1 0.397 0.392 0.386 0.379 0.374 0.366 

- 0.5 0.382 0.378 0.375 0.371 0.368 0.366 

rim 

0.25 0.373 0.371 0.369 0.368 0.366 0.366 

KOab 

0.1 0.367 0.367 0.366 0.366 0.366 0.366 

1 0.397 0.392 0.386 0.379 0.374 0.366 

0.5 0.300 0.293 0.286 0.279 0.274 0.269 

0.25 0.217 0.209 0.203 0.198 0.194 0.190 

KOCd(ll? 

- 

KW 

0.1 0.136 0.130 0.126 0.124 0.124 0.123 

56 
Table 22: The interaction between two cracks on the normalized stress 

intensity factor at the deepst penetration point of a semielliptic surface crack under uniform bending or tension 

(Material 2) 

B-A D-C a=- e=- d=C-B, "1 

2' 2' h 

B+A D+C Lo 

YOob = 7 YOc,. = 79 7 = 0*4 

- + 0.1 0.25 0.5 1 2 00 

d 

C 

- 

a 

1 0.444 0.437 0.430 0.423 0.417 0.409 

- 0.5 0.430 0.425 0.420 0.416 0.412 0.409 

KOD 

0.25 0.419 0.416 0.414 0.412 0.410 0.409 

0.1 0.412 0.411 0.410 0.410 0.410 0.409 

~~ 

1 0.444 0.437 0.430 0.423 0.417 0.409 

- 0.5 0.298 0.287 0.277 0.268 0.262 0.255 

Km 

0.25 0.171 0.158 0.148 0.140 0.135 0.130 

KOCd(B' 

0.1 0.054 0.043 0.036 0.032 0.029 0.027 

1 0.505 0.500 0.494 0.488 0.483 0.477 

VI - 0.5 0.494 0.490 0.486 0.482 0.479 0.477 KOab 

-_ KW 

0.25 0.485 0.482 0.480 0.479 0.478 0.477 

0.1 0.479 0.478 0.477 0.477 0.477 0.477 

1 0.505 0.500 0.494 0.488 0.483 0.477 

0.5 0.380 0.370 0.362 0.355 0.350 0.345 

0.25 0.270 0.260 0.252 0.245 0.241 0.237 

KOcd(q 

- 

KOD 

0.1 0.168 0.160 0.154 0.150 0.148 0.146 

57 
Table 23: The interaction between two cracks on the normalized stress 

intensity factors at the deepst penetration point of a semielliptic surface crack under uniform bending or tension 

( Material 3) 

B-A D-C 

a =-, c = - ,d=C-B, 2 

a 

-= 1 

h 

- 0.1 0.25 0.5 1 2 00 

d 

a 

c 

- 

a 

1 0.554 0.548 0.542 0.535 0.530 0.522 

- 0.5 0.542 0.537 0.533 0.529 0.525 0.522 

K0.25 0.532 0.529 0.527 0.524 0.523 0.522 

KOaJB’ 

0.1 0.525 0.524 0.523 0.522 0.522 0.522 

1 0.554 0.548 0.542 0.535 0.530 0.522 

KOcd(B) * 

- 0.5 0.403 0.393 0.383 0.374 0.367 0.361 

_. Ka 

0.25 0.256 0.242 0.231 0.223 0.217 212 

0.1 0.103 0.091 0.083 0.078 0.074 0.072 

1 0.600 0.596 0.591 0.585 0.581 0.574 

- 0.5 0.591 0.587 0.583 0.580 0.577 0.574 

K= 

Ko*Jq 

._ 

0.25 0.582 0.580 0.578 0.576 0.575 0.574 

0.1 0.576 0.576 0.575 0.575 0.575 0.574 

1 0.600 0.596 0.591 0.585 0.581 0.574 

- 0.5 0.470 0.461 0.453 0.446 0.440 0.435 

K0.25 0.343 0.332 0.324 0.316 0.312 0.308 

KOcd(T) 

0.1 0.21 1 0.202 0.196 0.192 0.189 0.187 

58 
Table 24: The effect of Lo/h on the normalized stress intensity factors at 

the maximum penetration point of two identical collinear semielliptic surface cracks under uniform tension or bending 

(Material 1) 

LO 

- -0.2 h 0.4 0.5 0.6 0.8 

0.1 0.832 0.488 0.313 0.165 -0.005 

0.25 0.829 0.481 0.306 0.159 -0.006 

0.5 0.826 0.470 0.299 0.153 -0.008 

- 1 0.821 0.465 0.290 0.147 -0.009 

KW 

2 0.816 0.456 0.283 0.141 -0.010 

00 0.811 0.446 0.272 0.135 -0.01 1 

KOab(B) 

0.1 0.842 0.545 0.397 0.270 0.0755 

0.25 0.840 0.540 0.392 0.266 0.0764 

0.5 0.837 0.533 0.386 0.261 0.0753 

- 1 0.833 0.526 0.379 0.256 0.0742 

KOo 

2 0.828 0.518 0.374 0.252 0.0733 

KOab(q 

00 0.823 0.511 0.366 0.247 0.0725 

59 
Table 25: The effect of Lo/h on the normalized stress intensity factors at 

the maximum penetration point of two identical collinear semielliptic surface cracks under uniform tension or bending 

(Material 2) 

0.4 0.6 0.8 LO 

- -0.2 h 

0.1 0.815 0.444 0.121 -0.021 

0.25 0.812 0.437 0.11 5 -0.0224 

0.5 0.809 0.430 0.1 10 -0.0235 

- 1 0.806 0.423 0.105 -0.0245 

KW 

2 0.802 0.417 0.101 -0.0253 

00 0.798 0.409 0.097 -0.0260 

0.1 0.826 0.505 0.234 0.0622 

0.25 0.823 0.500 0.230 0.0611 

0.5 11.820 0.494 0.226 0.0602 

- 1 0.817 0.488 0.222 0.0592 

Km 

2 0.814 0.483 0.219 0.0585 

00 0.810 0.477 0.215 0.0578 

60 
Table 26: The effect of Lo/h on the normalized stress intensity factors at 

the maximum penetration point of two identical collinear semielliptic surface cracks under uniform tension or bending 

(Material 3) 

B+A B-A 

YOab = 7 , a=- , d=C-B, :=I, C "1 h 

0.4 0.6 0.8 

LO 

- 4.2 h 

0.1 0.873 0.554 0.201 -0.0026 

0.25 0.871 0.548 0.194 -0.0045 

0.5 0.869 0.542 0.189 -0.0062 

- 1 0.866 0.535 0.183 -0.0078 

K2 0.864 0.530 0.178 -0.009 

a3 0.861 0.522 0.172 -0.0103 

0.1 0.880 0.600 0.301 0.0822 

0.25 0.878 0.596 0.297 0.081 

0.5 0.876 V.3YI 0.292 0.0?93 

- 1 0.874 0.585 0.288 0.080 

K2 0.872 0.581 0.283 0.077 

00 0.869 0.574 0.279 0.076 

(TI KOab 

61 
Table 27: The effect of the thickness ratio on the stress intensity 

factor in a plate under uniform bending, containing a 

single crack, u,=6Mo/h2 (Material 4 and Material 5) 

a 

h 

- Mat.1 Mat.4 Mat.5 

0.05 

0.125 

2 

4 

8 

0.9885 

0.9559 

0.8992 

0.8195 

0.7477 

0.7003 

0.6701 

0.6521 

0.9894 

0.9591 

0.9059 

0.8293 

0.7571 

0.7015 

0.6733 

0.5916 

0.9878 

0.9536 

0.8950 

0.8144 

0.7432 

0.691 1 

0.6675 

0.5864 

62 
Table 28: Stress intensity factors at the maximum penetration point of 

a semi-elliptic surface crack in an infinite plate under uniform 

tension or pure bending (Material 4 and Material 5) 

0.5 

1 

0.710 

0.809 

0.303 

0.441 

0.049 

0.130 

-0.029 

-0.013 

2 0.872 

0.923 

0.946 

0.575 

0.704 

0.776 

0.233 0.015 

0.365 

0.458 

0.062 

0.101 

4 

6 

Mat.4 

0.5 

1 

0.729 

0.820 

0.387 

0.506 

0.173 

0.244 

0.049 

0.071 

2 0.881 0.623 0.332 0.102 

KW 

4 

6 

0.928 

0.949 

0.736 

0.799 

0.445 

0.523 

0.146 

0.182 

0.5 

1 

0.697 

0.797 

0.290 

0.424 

0.558 ' 

0.044 

0.121 

0.221 

-0.0293 

-0.0141 

- 

Klb 

KW 

2 0.864 0.012 I 

0.918 

n.943 

0.691 

0.766 

0.351 

0.444 

0.057 

0.095 

4 

6 

Mat.5 

0.5 

1 

0.717 

0.809 

0.375 

0.491 

0.169 

0.236 

0.048 

0.070 

2 0.873 0.608 0.322 

0.433 

0.512 

0.099 

0.142 

0.176 

KW 

4 

6 

0.923 

0.945 

0.725 

0.790 

63 
Y / 

Figure 1: Notation for moment and transverse shear resultants. 

64 
I 

Y 

Figure 2: Notation for the part-through surface crack. 

65 
Y 

'I 

I 

. 

t 

I 

Figure 3: Geometry and notation for two collinear cracks. 

66 
Mat.1 

Mat.2 

Mat.3 

-.-.- 

-- - - 

I *# I 

I 

I 

i 

1 .o 

0.9 

kl 

.. 

0.7 

0.6 

- 

I. 

/* 

/- 

I 

1 

0.2 0.6 1 .o 0.6 0.2 0. 

h/a . a/h 

Figure 4: Stress intensity factor in a cracked plate under uniform bending. 

67 
I. 

I. 

1.1 

1.0 

0.9 

0.8 

0.7 

I 

c/a 1. 0.5 0.25 0.1 

klBIUbJO 1 2 3 4 

1. 2. 3. 4. 

Figure 5: Stress intensity factors in a plate containing two collinear 

through cracks, ( a/h= 1 , Fig. 3, Mat.1). 

68 
kl 

1.2 

1.0 

0.8 

0.6 

0.4 

- 

7 

s 

2. 3. 4. 5. 1. 

Figure 6: Stress intensity factors in a plate containing two collinear 

throueh cracks. a/h = 1 . Fig. 3. Mat.1). 

69 
1.0 

-. 0.5 

at.2 

at.3 

2. 4. 6. 8. 

Figure 7: Stress intensity factor at the maximum penetration point of a 

semi-elliptic surface crack in an infinite plate under pure bending. 

70 

f 
.. .. 

.. 

4 i 1.0 

! 

I 

I 

I 

, 

I 

I 

I 

I 

I 

I 

I 

I 

I 

0.5 

J 

I 

0. ' 

.. 

' - h4a1.2 

-- - Ma13 

a/h 

Figure 8: Stress intensity factor at the maximum penetration 

point of a semi-elliptic surface crack in an infinite 

plate under uniform tension. 

-3 

I I 

- 2 

I 

6. 8. 2. 4. 

71 
I 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

13. 

References 

J.R.Rice and N.Levy, “The part-through crack in an elastic plate”, 

J. Appl. Mech., Vol. 39, Trans. ASME, pp185-194. 

F.Delale and F.Erdogan, “Line spring model for surface cracks in a 

Reissner plate”, Int. J. Eng. Sci., Vo1.19, pp.1331, 1981. 

F.Erdogan and H.Boduroglu, “Surface cracks in a plate of finite 

width under extension or bending,’ Theoretical and Applied Fracture 

Mechanics, Vol. 1, 1985. 

F.Erdogan, “Theoretical and experimental study of fracture in 

pipelines containing circumferential flaws”, Final Report under 

contract DOT-RC-82007, DOT-RSPA-DMA-50/83/3, Sept. 1982 

E.Reissner, “The effect of transverse shear deformation on the 

bending of elastic plate”, J. Appl. Mech, V01.12, Trans. ASME 

ppA69-A77, 1945. 

E.Reissner, “On bending of elastic plate”, Quart. Appl. Math, Vol. 5, 

pp. 55-68, 1947-1948. 

S.Timoshenko, “Theory of plate and shells” pp165-171, McGRAW-Hill 

book company, 1959. 

S.J.Medwadowski, “A refined theory of elastic, orthotropic plate”, 

J. Appl. Mech, Vo1.80, Trans. ASME. pp437-443, 1958. 

F.Erdogan, “Mixed boundary value problems in mechanics”, 

Mechanics Today, S.Nemat-Nasser, ed., Vol. 4, ppl, Pergamon Press, 

Oxford, 1978. 

M.B.Civelek and F.Erdogan, “Elastic-plastic problem for a plate with 

a part-through crack under extension and bending”, Int. J. of 

Fracture, Vol. 20, pp33-40, 1982. 

F.Erdogan, “The line spring model”, Interim Report, Lehigh 

University. 

A.C.Kaya and F.Erdogan, “Stress intensity factors and COD in an 

orthotropic strip”, Int. J. of Fracture, Vol. 16, pp171, 1980. 

G.C.Sih and Liebowitz, “Mathematical theories of brittle fracture”, 

Fracture - an advanced treatise, H.Liebowitz, ed., Vol. 2, ppll0 and 

127. 

72 
14. R.F.S.Hearmon, 'An introduction to applied anisotropic elasticity", 

Oxford University press, p38, 1961. 

73 
I 

t 

Appendix A 

Definition of constants A. J and B. J 

h2 '55 A =--- 

2 lo S6, ' 

h3 2n A =A ---(---S ) 4 7- 12n S6, 12 

h2'44 A =--, 

5 10S6, 

B, = -A,, 

B, = -2A4, 

B3 = -A,, 

B4 = - A,A5, 

B5 = - A4A5 - A3A, - A, A, , 

B, = -A4A, - A2A4 - A,A, , 

B, = - A2A, . 

74 
Appendix B 

Expressions for R. 3 

( j = 1, 2, 3 for z > 0) 

where 

D(a) = (Q2a4 + Q1a2 lA2A3 ( A3 - A, + A1A3 ( A1 - A3) + A1A2 ( A, - A1 ) ] 

m, are roots of equation (13) and Ai= mi2. 

Q1=Q3=P5P1> 

75 
h3 1 

p =-- 

'66 

h5 '55 

P4 = --s12, 120K s66 

h3 

p!i = ,,,'22 ' 

h3 1 h3 

p =---+- 

6 SSs 12~'~~' 

h2 h2 '44 

P,=--S s +--, 

lo '66 loti l2 44 

h2 h2 '55 

p=-s ' ---. 

lo '66 

8 ioK 55 12 

76