Analytical estimation of stress intensity factors in patched cracked plates

Engineering Fracture Mechanics Vol. 21. No. 3, pp. 479-494, 1985 0013-7944/85 $3.00 + .130 Printed in the U.S.A. �9 1985 Pergamon Press Ltd.

ANALYTICAL ESTIMATION OF STRESS INTENSITY FACTORS IN PATCHED CRACKED PLATES

RAMESH CtlANDRA and M. V. V. MURTHY Structures Division, NAL, Bangalore, India

and

T. S. RAMAMURTHY and A. K. RAO Aero. Engng Dept., IISc, Bangalore, India

Abstract--The fatigue and fracture performance of a cracked plate can be substantially improved by providing patches as reinforcements. The effectiveness of the patches is related to the reduction they cause in the stress intensity factor (SIF) of the crack. So, for reliable design, one needs an accurate evaluation of the SIF in terms of the crack, patch and adhesive parameters. In this investigation, a centrally cracked large plate with a pair of symmetric bonded narrow patches, oriented normally to the crack line, is analysed by a continuum approach. The narrow patches are treated as transversely flexible line members. The formulation leads to an integral equation which is solved numerically using point collocation. The convergence is rapid. It is found that substantial reductions in SIF are possible with practicable patch dimensions and locations. The patch is more effective when placed on the crack than ahead of the crack. The present analysis indicates that a little distance inwards of the crack tip, not the crack tip itself, is the ideal location, for the patch.

N O T A T I O N

2a crack length 2b width of cracked plate or sheet E Young's modulus G shear modulus

Im imaginary part of complex function Kr stress intensity factor of cracked sheet due to shear stress distribution acting between patch and sheet K stress intensity factor of cracked sheet with patch, subjected to uniform tensile load 21 length of patch

Re real part of complex function t thickness v displacement in y-direction w width of patch z x + i y

Zo xo + i Yo a EptpwlEstsa tr applied uniform stress v Poisson's ratio.

Subscripts a adhesive p patch s sheet.

I N T R O D U C T I O N

THE 'FAIL SAFE' c o n c e p t is w i d e l y used in the des ign o f a i rc ra f t s t r uc tu r a l c o m p o n e n t s . T h e des ign o f such ' d a m a g e t o l e r a n t ' s t r uc tu r e s is g o v e r n e d by de t a i l ed s p e c i f i c a t i o n s [1, 2], w h i c h a re i n t ended to e n s u r e tha t a spec i f i ed fa t igue c r a c k will not g r o w to c r i t i ca l p r o p o r t i o n s in a pe r iod b e t w e e n s u c c e s s i v e rou t i ne i n spec t ions .

Fa t i gue c r a c k e d c o m p o n e n t s a r e o f ten r e p a i r e d in se rv ice . S t a n d a r d r e p a i r s c h e m e s n o r m a l l y invo lve s t r e n g t h e n i n g the c o m p o n e n t b y c o n n e c t i n g a r e in fo rc ing m e m b e r b y m e a n s o f bo l t s o r r ive t s and t h e r e b y r e d u c i n g the c r a c k tip s t r ess in t ens i ty f ac to r s .

R e c e n t t e c h n o l o g i c a l a d v a n c e s in f ibre r e in fo rced c o m p o s i t e m a t e r i a l s and a d h e s i v e b o n d i n g have led to the d e v e l o p m e n t o f e f f ic ien t r epa i r s c h e m e s us ing these . In such r e p a i r s , (a) the load t r ans fe r b e t w e e n c o m p o n e n t and r e i n f o r c e m e n t is a f fec ted wi th min ima l s t ress c o n c e n - t ra t ion e f fec t s , (b) the p r o p e r t i e s and g e o m e t r y o f the r e i n f o r c e m e n t c a n be t a i lo red to sui t the

479

480 R. CHANDRA et al.

particular application and (c) the composite repair patch does not add significantly to the weight of the component. Boron-epoxy, boron-aluminum and carbon-epoxy have been used in some repair schemes for aircraft structures [3, 4].

An efficient design of the reinforcement (patch) needs estimation of the resulting stress intensity factor (SIF) at the crack tip in the patched panel.

There have been numerous attempts [5-8] to compute SIF of patched cracked plates. These are based on the finite element technique using special elements with displacement [9] or hybrid formulation [10].

In this investigation, stress intensity factors in a large plate with a central crack, symmet- ricaUy reinforced with patches on both sides, are estimated using a continuum approach. The reinforcement is represented as a transversely flexible line member. Hence, the problem is reduced to one of a cracked plate with distributed applied shears, unknown ab initio. This shear force distribution is determined as follows: the displacement field in the cracked plate, due to combination of external uniaxial tensile loading and distributed shear forces caused by the patches, is obtained from elasticity solutions. The one-dimensional displacement fields in the reinforcements are directly evaluated in terms of the shear forces. Also, the differences between displacements in the sheet and patch are accounted for by the shear flexibility of the adhesive. Thus an integral equation in shear stress results. This equation is numerically solved by assuming a polynomial distribution in shear stress and employing a point collocation technique. The S1F in the cracked plate due to the distributed shear forces is then computed by using point load solutions (SIF in centrally cracked infinite plate due to two symmetric pairs of equal and opposite unit point forces). The total SIF in the patched plate is computed by the direct addition of the SIF's due to the external loading and the distributed shear forces.

2. MATHEMATICAL DEVELOPMENT

Figure I(a) shows the configuration of a central crack in a large plate [(b/a) > 10], patched symmetrically. In Fig. l(b) the patches are replaced by unknown shear stress distributions along the patch central lines. The stress intensity factor for the patched and loaded plate, Fig. l(b) is given as

K = crN/('tra) + g r (z)

O"

t I T 1 ,Y

W ~ -p,o 1 2 b

O"

b / a ) I0

L . ~X

O"

1 1 I I I Y ]i t

t " r ( y ) r ( y ) t t I t t |

t

t t

O"

Fig. I. (a) Configuration of patched plate. (b) Cracked plate with shear due to patch.

Analy t i ca l e s t imat ion of s tress intens i ty factors 481

where

K~ = w f : "~(Xo, yo)dp(Xo, 5'0) dyo

�9 r(Xo, Yo) is the shear stress at point (Xo, yo) in the sheet and +(Xo, yo) is the stress intensity factor in the sheet due to the set offour unit forces acting at (-+-Xo, -yo) . qb(Xo, yo ) was derived by Sneddon and Tweed [11] employing Fourier transforms and the results were obtained in the form of integrals, which are not convenient for the present purpose. Hence ~b is now rederived in closed form using complex potentials (see Appendix A)

2 / ( a ) [ i m { ( z Z o _ a Z ) _ o . 5 } 1 + v ] ~b(Xo, Yo) = - t~ ~ 5'0 R e {Zo(Z~ - aZ) - 1.5} . (2)

Figure 2 shows the comparison of the results of the two techniques.

2.1 Shear stress estimation Under the assumption of uniform direct stresses across the thicknesses of sheet and patch,

the shear stress in the adhesive is given by the formula

aa T = ~ (v~ -- vO. (3)

Considering that the load transfer between patch and sheet is by surface shears at the interfaces with the adhesive and the other faces are shear free, it is closer to reality to model a linear distribution of shear from zero at its free surface (or mid surface in a symmetric reinforcement) to a maximum at the adhesive interface as in Fig. 3 [6]. This yields the following expression

7 - 0

6 . 0

5 - 0

4 . 0

a.

,,e ~ . 0

~r 2 . 0

I .O

- - E Q U A T I O N ( 2 )

SNEDDON [ '11] Xolo

0 0 ' 2 5

A 0 - 5 0

r~ I - O 0

r l

o I. I I I i i I I 1 i f 0 0"5 I" 0 I '5 2-0 2"5 3"0 :5"5 4-0 4"5 5 0

Y o / 0

Fig. 2. C o m p a r i s o n o f S I F s in large plate with central crack subjected to t w o s y m m e t r i c pairs o f equal , oppos i t e and equidis tant point forces .

482

I I ! ] fbd OF ADHESIVE

ztfbd OF CRACKED PLATE

HALF OF JOINTICRACKED PLATE~PATCH AND ADHESIVE)

R. CHANDRA et al.

-"i~Y h- o-y ~ ~-y +~.y

"z

ADHESIVE ELEMENT

Fig. 3. (a) Schematic diagram of patched plate. (b) Variation of shear across the thickness of the patched plate.

for the adhesive shear stress,

where

Ga 1 = - - - ( V p - v~) (4)

ta g

1 Ga/ta 3 GJta g = l + - - - + - - - 4 G~/ts 8 Gp/tp

As g is ahvays greater than one, use of relation (4) is equivalent to increasing the shear flexibility of the adhesive layer. Consequently, the shear peaks are lowered and the variation is smooth- ened.

v~ is given as

vs(xo, yo) = w[f:C~v(xo, Yo, y)r(xo, Yo) dyo]r=,.o +C~s(xo, yo)cr (5)

where Csv(Xo, yo, y) is the v displacement in the sheet at any point (Xo, y) due to unit point forces acting at (-+Xo, ---3'0) points. Cs~(Xo, yo) is the v displacement in the sheet at any p o i n t (Xo, yo) due to applied uniform stress, or. C~s from Ref. [12] is as follows:

1 [2~/(rlr2) sin 01 + 02 css = e-q L - - T - Yo - (1 + v) Yo

~/(rnr2)

( 0t + 02 0n + 02 ~r (6) • XoC0S-- T - + yos in- - - -T-- -

where

rt = [ ( x o - a) z +y~]~ r2 = [(Xo + a) z +3'2] 0.5

01 = tan-l(yo/(Xo - a)); 02 = tan-~(yo/(Xo + a)).

Analytical estimation of stress intensity factors 483

Y P P t

( -x o ,Yo ) ( Xo,Y o)

b-2o --~ ~"

(- xa ,-Yo ) (xo,-Yo)

T P P

2b

b/o ) I0

--p11.- +

NO CRACK

Fig. 4. (a) Large plate with central crack with a set of four point forces. (b) Large plate with a set of four point forces. (c) Large plate with central crack subjected to loading on crack faces.

2.1.1 De te rmh ta t ion o fCsp . To determine Csp, consider the displacement field in the cracked sheet due to four point forces as shown in Fig. 4. This is obtained by superposing the solutions of Figs. 4(b) and (c).

Vb, the y displacement at point (.,Co, y) in an infinite plate due to unit forces at points (-+Xo, -+Yo) (Fig. 4), is derived in Appendix B to yield

vb(xo, Yo, Y) = (3 v)(l + v) cxzta 2 - 2~ 2 + a])

I n 87rE~ts 2 2 0/-4(0/- 3 + 0/.24)

(1 "b V)2 [ )3'nOt 2 ] + + "

(7)

vc, the y displacement at any point (Xo, y) in an infinite plate with a central crack due to distributed stress %. acting over the crack surfaces (fig. 4(b)) is obtained as follows: %.(Xo, Yo, ~, 0) due to unit forces at (-Xo, ---Yo) as derived in Appendix B is

l + v ffy(XO, YO, ~, 0) = - - [fs(X0, YO, ~, 0) + f6(Xo, Yo, ~, 0)] (8)

2~rts

Y!

pI Cr,e)

Xo~Y o X!

O" ~X

Fig. 5. Large plate with point force.

484

where

f5 - - -

f6 ~--" - -

R. CHANDRA et al.

3 + v yo 1 + v ( 6 - X o ) 2 + y2

3 + v Yo I + v ( 6 + X o ) 2 + y ~

2yo(~ 'Xo) 2 [ (~ - Xo) 2 + y?~]~

2yo(~ + Xo) 2

[ (6 + Xo) 2 § y2o12"

The v displacement at point (Xo, y) due to pairs of unit forces acting on crack surfaces (Fig. 6) [13] is given as

where

I v(xo, y, 0 = - - (f7 - f8) (9)

'rrEs ts

f7 = In[C3 + 2V/(C3(CI + C2)) + 2C2]

- In[C3 - 2"k/(C3(Ci + C2)) + 2C2]

f8 - (1 + v)~r Y [ C4~//(C2-CI)XO-CSy~//(CI-t-C2)']" C--~ -q- -4x2y -''~

C i : �89 -~ y2 x~) ; C 2 ~-- ( C I ..Jl- x ~ ) " ) "

C3 = a 2 - ~2; C4 = x~ + y2 _ ~2

Cs = x ~ + y ~ + ~z., C 6 = . r ? ~ - y ' - - ~ .

The displacement Vc due to try [as defined in eqn (8)] acting over the crack surfaces is obtained as

l + v vc - 27r2Ests l(xo, to, y) (10)

where

f o a l ( xo , Yo, Y) = ( f5 + f 6 ) ( f 7 - f s ) d~

Csp is obtained by adding relations (7) and (10) as follows

Cso ( 3 - v ) ( l + v ) [ 8(I + v ) 4 ] = 8~,~t~ - g l (3 - v) gz + "rr(3 - v-"~ I (1 I)

,Y

i, ~,o,t x P P

Fig. 6. Large plate with central crack with point forces acting on the crack faces.

Analytical es t imat ion of s t ress intensity factors 485

where

2 2 "~ 0[2(0[2 + 0[~) gl = In 2 2 0[-4(0[ 3 + Of 2)

Y)'o0[~ g2 = (0[2 + 0122)(0[~ + e~4 2)

gl is singular at y = yo and it is desirable to isolate the singular part. Thus

where

Using relations (11) and (12)

gn = 2 In()' - Yo) + g3

where

g3 = In 2 2 a4(a 3 + a4 2)

(12)

C s p ( 3 - v ) ( l + v ) [ = 8(1 + v) g2 + 4 ] . I " 8~-sst~ - g 3 3 - v ,rr(3 v) "

2.1.2 Estimation o f displacement Vp in the patch due to shear stress distribution. From the patch equilibrium equation

"r = tp(Yy.y ( 14 )

where cry is the normal stress in the patch in the y-direction. Integrating eqn (14) and enforcing the zero normal stress condition at y = l (free end)

1 f/' tr s = -- "r dy (15) tp

Using s t ress-s t ra in and s t ra in-displacement relations

Is vr"Y = Eptp "r dy. (16)

Integrating eqn (16) and enforcing the symmetric (zero displacement) condition at y = 0

vp = Eptp .-r @2. (17)

2.1.3 Governing equation in shear stress and its solution. Using relations (4), (5) and (17), the following integral equation is obtained

foYLl,rdy2 EplpW I f to Csp,Tdy] fptpta - G------~'gr - EotpcrCs~ = 0. (18)

Equation (18) is solved by assuming a polynomial distribution for shear and employing a collocation technique.

(3 - v)(l + v) Csp = - 4~rEsts I n ( ) , - Yo) + C~pz (13)


Shear s t r e s s "r is a s s u m e d as follows

n

= E A j / - ' . j = l

(19)

Using relation (19) in eqn (18), the following relation r e s u l t s

n

E Z j f j = G o C s s j = l

(20)

where

1 Ga G - g ta

+ , ] "

fJ = SE--p-Tp [d: + "1 1@o - y~-'

(3 - v)(1 + v) b = 4~rEsts Ij, + Ij2

b, = f : ln(y - yo)YJo - i dyo

fo I j - I /J'2 = Csp2Yo dyo.

- w G I j ( y o )

E E

I J

I0

- Y (mm)

I0 8 6 4 2 0 t 1 i I I 0

f I I I I v

0 Z 4 6 "13 I0

IO

Y (mm)

Fig. 7. Shear stress distribution along the patch length in photoelastic specimen.

j - t ( j - 1)! Ij, = - E r ! ( ) m i - r)!

r = O

I - - r r.>,'_+'. ,,_ y _y,-+ t 1 ( - l ) r ) u - L r + 1 (,'+ l)2J

"i_~10 ( j - - 1 ) ! yj_l_r [ ( l - - y)r+i(! -- y) - r ! ( . ] - - -1 - - r ) ! r + 1

(! - y ) r+ l ]

j

Collocating eqn (20) at n values of yo,

[al{A} = {F} ( 21 )

1.5

where

au = f.i(Yi)

{F} = Gcr{Css(y t )Cs~(yz) . . . Css(y,,} r

{Z} = { A I A z A s . . . A,,} r.

The following collocation scheme is used

! yi = ( i - 0.5)(n - 0 .5) ' i = 1 , 2 , 3 . . . n . (22)

{A} is obtained by solving eqn (21). Thus the shear stress distribution is determined from eqn (19) and K from eqn (I).

1.0

~J

E E

Z

P I 0"5

0 4 8

X 0 =9mm, I =20ram

t o -O . lO I6mm , ~ -0 "017


I/, is singular at y = Yo; its numerical evaluation poses convergence problems. Hence lj, is evaluated in closed form and is given as

/ 12 16 20

ytmm)

Fig. 8. Shear stress distribution in a typical patched plate.


=3

E E

Z

I

0-75

0-70

0 - 6 5

0 ' 6 0

o o - - -o - - - -o

O m m

l _ . = �9 O~ : 0 - 0 1 7 ~ 10 0 |OI6mm

0 - 5 5 I I I 1 l I 3 ,5 7 9 II 13 15

COLLOCATION POINTS

Fig. 9. Variation of K, with collocation points in a typical patched plate.

v "

1.0

(15

XO=9 m[~

t o = 0"I016 mm

[3 O" 104

~ o O- 209 ~ x s 0 - 3 1 4

,&

0 . O t I I

O I.O 2.0 5.0

I /o Fig. 10. Variation of SIF with patch length for various patch stiffness ratios.


3. RESULTS AND DISCUSSIONS

To validate the technique developed in this paper , let us consider a thin rectangular sheet o f dimensions 75 • 340 • 2.96 mm, made ofphotoe las t i c material. The sheet contains a central crack of 25.5 m m length which is symmetr ical ly patched on both sides by two photoelast ic strips. The dimensions of the patch are: 20 • 2.5 • 1.71 mm. The 1.71 m m thick patches are bonded to the c racked sheet with a room tempera ture curing adhes ive of 0.01 mm thickness. Young ' s moduli and Poisson ' s ratios of both patch and sheet are 3100 N mm -2 and 0.36, respectively. Shear modulus o f sheet, patch and adhesive equals 1140 N m m -2. The patch centre line is at a distance of 6.88 m m from the crack centre. The cracked sheet is subjected to a far field uniform stress of 3.61 N m m -2 . The shear stress distribution obtained is shown in Fig. 7. The theoretical value of S IF for this configuration is 15.22 N m m - l 5 whereas the exper imental value as obtained by employing the photoelast ic technique [14], is 13.8 N m m - 1.5. The difference is within 10%.

Fur ther numerical work is carried out for an aluminium alloy plate with c a r b o n - e p o x y patch with the following data:

Es = 71,000 N mm -2,

Gs = 27,300 N m m -2

Ep = 208,000 N mm -2,

t o = 0.127-0.762 m m

Ga = 965 N m m -2 ,

w = 2-10 mm,

Xo = 5 -19 mm,

v = 0.3

Gp = 5000 N mm -2

ta = 0.1016 and 0.2032 m m

tr = 0.689 N m m -2

a = 1 9 m m

! = 5-30 mm.

i= E

z

v

I

1.0

0 . 5

I = 2 0 m m ~ X o =gram CC

0 0 . 2 0 9

O 0-.'514

0 . 5 2 5

I I / f

o 4. 8 12 tG 2 0

Y ( m m )

Fig. I I. Shear stress distribution in patched plate for various stiffness ratios.

490 R. C H A N D R A et al.

Table I. Adhes ive shear distribution for various collocation points (CP)

yll

CP 0 0.2 0.4 0.6 0.8 !.0

6 - 1.169 - 0 . 1 6 4 - 0 . 0 1 9 - 0 . 0 0 2 - 0 . 0 2 7 -0 .138 7 - 1.267 - 0 . 1 5 7 - 0 . 0 2 0 - 0 . 0 0 3 - 0 . 0 2 3 - 0 . 3 0 8 8 - 1.304 - 0 . 1 5 6 - 0 . 0 1 9 - 0 . 0 0 3 - 0 . 0 2 4 - 0 . 2 3 2 9 - 1.327 - 0 . 1 5 6 - 0 . 0 1 9 - 0 . 0 0 3 - 0 . 0 2 4 - 0 . 3 1 8

10 - 1.331 - 0 . 1 5 6 - 0 . 0 2 0 -0 .003 - 0 . 0 2 4 - 0 . 2 9 3 I 1 - 1.329 - 0 . 1 5 6 - 0 . 0 2 0 -0 .003 - 0 . 0 2 4 - 0 . 2 9 2

la

I.O

0 - 5

O 0

I I I 0.5 I 0 1.5

Fig. 12. SIF o f patched plate for various values of s t i ffness ratios.

I 2.0

1"5

I 2 '5

I - 0

E E

z

~" 0 " 5 I

I = 2 0 r a m C~ : 0 - 1 0 4

t o l m m ) 0 . 1 0 1 6

. . . . 0 . 2 0 3 2

I

0 4 8

y(mml

j•/j/l 2 16 2 0

Fig. 13. Shear s t ress distribution for various adhes ive th icknesses .


b v '

0.8

0.6

0.4

02.

0 . 0 l = T t I t I

0 4 8 t2 t6 2 0

O �9 19mm

I = 2 0 r a m

t o = 0.1016 mm

CC 0 0.017

A 0 .087

n 0.520

Xo (ram)

Fig. 14. Variation of SIF of patched plate with patch position for various stiffness ratios.

~ N O PATCH

~ ' ~ PATCH CENTRE LINE

z

0 t I I 0 0-5 1-0 I-5 2.0

a / X o

Fig. 15. Variation of SIF with crack length for fixed patch position.

In order to establish the optimum collocation points from the convergence view point, shear stresses and SIF(Kr) are estimated for a typical case with the number of collocation points varying from 3 to 15. The convergence trend is good. The variation of values beyond nine collocation points is negligible. Table I and Fig. 8 show some details of shear stress distribution and convergence. Figure 9 shows the convergence of SIF(Kr) with the number of collocation points.

Influence ofpatch length on SIF(K) is shown in Fig. 10. It is seen that the value of K virtually stabilizes at l/a = I. Hence, all further numerical work is carried out for a 40 mm patch length (l/a = 1.05).

Figure I I illustrates the effect of patch axial stiffness ratio, a, on shear stress distribution. As expected, increase in patch stiffness results in increase of shear stress at the far end. How- ever, the shear stress at the crack end is reduced by increasing patch stiffness. Figure 12 shows the effect of patch stiffness on SIF. Increasing stiffness ratio reduces SIF and it reaches an asymptotic value for a = 2. It is apparent that substantial reduction in SIF can be achieved with relatively small patch stiffnesses. Shear peaks at crack and far ends are naturally reduced with increasing the adhesive flexibility (tJG~). This is shown in Fig. 13.

Figures 14 and 15 indicate the influence of patch position on SIF. There seems to be an optimum patch position on the crack for which SIF is minimum. Figure 15 shows SIF vs crack


length for a fixed patch position. Such representation is useful to study crack growth in patched plates.

4. CONCLUSIONS

The main intention of the present work was to establish the methodology for continuum analysis of patched cracks. Although only line reinforcement is considered in the present work, the method can easily be extended to a two-dimensional patch by idealizing it as a series of line reinforcements. Such an idealization should be reasonably good because shear forces par- allel to thecrack are known to produce negligible changes in SIF. However, i fa more accurate analysis is required, a two-dimensional model can be used for the patch. Such an analysis may be necessary to establish the actual benefits possible by covering the crack tip with a patch.

Although the problem of patched cracks has been treated in the past extensively through finite element methods, it appears that this is the first time that the problem is examined by a continuum approach. The advantages of this technique over finite element techniques are: (l) functional representation of shear stresses in the adhesive is feasible; (2) parametric study with less computational effort is possible.

From the parametric study carried out the following conclusions are drawn: (1) There seems to be an optimum patch length beyond which there is no significant reduction

in SIF. (2) There seems to be an optimum position of patch with respect to crack for which the

reduction in SIF is maximum.

Acknowledgements--Financial support for this work was provided by Aeronautics Research and Development Board. Thanks are due to Mr Jayant Sonavane for his computational assistance. Thanks are also due to Mr K. Purushothaman for his help in preparing the manuscript.

REFERENCES

[I] N. N., Damage tolerance and fatigue evaluation of structure FAA, 14 FCR Part 25, 571 (1977). [2] N. N., Airplane damage tolerance requirements. MIL-Spec., 83444, USAF (1974). [3] A. A. Baker and M. M. Hutchinson, Fibre composite reinforcement of cracked aircraft structure. Tech. Memo.

ARL/Mat. 366, August (1976). [4] A. A. Baker, A summary of work on application of advanced fibre composite at ARL Australia, Composites 9(I),

1 1 - 1 6 (1978). [5] R. Jones and R. J. Callinan, On the use of special crack tip elements in cracked elastic sheets. Int. J. Fracture

13(1), 51-54 (1977). [6] R. Jones and R. J. Callinan, Finite element analysis of patched cracks. J. Sir. Mech. 7(2), 107-130 (1979). [71 R. Jones and R. J. Callinan, A design study on crack patching. Fibre Sci. Tech. 14, 99-111 (1981). [8] M. M. Ratwani, Analysis of cracked adhesively bonded laminated structures. AIAA J. 17(9), 988-994 (1979). [9] A. K. Rao, I. S. Raju and A. V. Krishna Murthy, A powerful hybrid method in finite element analysis. Int. J.

Num. Method Engng 3, 389-403 (1971). [10] P. Tong, T. H. H. Pian and S. J. Lastry, A hybrid element approach to crack problems in plane elasticity. Int.

J. Num. Method Engng 7, 297-308 (1973). [11] I. N. Sneddon and J. Tweed, The stress intensity factor for a Griffith crack in an elastic body in which body

forces are acting. Int. J. Fracture Mech. 3, 317-330 (1967). [12] H. M. Westergaard, Bearing pressures and cracks. J. Appl. Mech. 16(2), A49-A53 (1939). [13] G. R. Irwin, Analysis of stresses and strains near the end of a crack traversing a plate. Trans. Am. Soc. Mech.

Engrs, Series E, J. Appl, Mech. 24(3), 361-364 (1957). [14] A Subramanian and Ramesh Chandra, Stress intensity factors in plates with partially patched central crack, In

Commltnication with Experhnental Mechanics. [15] C. T. Wang, Applied Elasticity. McGraw-Hill, New York (1953).

APPENDIX A DETERMINATION OF SIF IN A LARGE PLATE WITH CENTRAL CRACK DUE TO TWO PAIRS OF POINT

FORCES

DUE "to a concentrated load P at (xo, Yo) (Fig. 5) the stress system is defined by the following Airy stress function

P = ~ R e [ i l n ( z - zo){(3 - v ) ( z - zo) + (1 + v ) ~ - zo)}] . ( A I )


%.(~, 0) as obtained by using eqn (Ai) is given as

~ry(~, 03 = ~ t In -- -- . (A23 - zo ( ( -" z~3 ~ (~- - ~o)

~ry(~_o) due to P at (xo, Yo) and - P at (Xo, -Yo) is

Pln(~___~lzo) I + V P y o R e ( ~ ) (A3) ~ y ( ~ , 0 ) = - 7 2 ~ t "

From Irwin's solution for the problem o f cracked surfaces subjected to two equal and opposi te point forces as shown in Fig. 6, SIF is given as

: _ t (a z _ ~2)o.5" (A4)

SIF due to Cry(~, 0) [as defined in eqn (A3)] is

J(o)s:o ' SIF = 2 ~,, a>.(~., 0) (a 2 _ ~2)o.~ d~.

Using relations (A3) and (A5), SIF due to four unit point forces acting at (-+xo, ---Yo) is given as

dp(xo, So) = - t [Im{(zo 2 - + [ ( - - ~ - ' ] Y o Re{zo(z~ - a : ' ) - ' " } i . (A6)

APPENDIX B STRESS ANALYSIS OF INFINITE PLATE WITH POINT FORCES

STATE Or stress at a point due to load P at point A in infinite plate (Fig. 5) in polar coordinates as given in Ref. [15] is indicated as

13" o - - _ _

O ' r o

3 + v l P sin 0

47rt r

I - - u l P sin 0 (BI)

4~t r

I - - v i P cos 0.

4wt r

The above relations are rewritten in xoq coordinates as follows:

O ' y I

O ' S l y I =

P sin 0

4wt r [I - v - 2(I + v) cos 20]

P sin 0

4"~t r - - [3 + v - 2(1 + v) cos 20]

P cos 0

4,-rt r - - [1 - v + 2(I + v) sin 20].

(B2)

Relations (B2) are rewrit ten in xy coordinates as below

(x - xo) 2 ] P if_. Y - Yo 1 - v - 2(i + V)(x - xo) 2 + (y - 3"o) 2

P Y - Y o [ 3 + v 2(' + (x--x~ ] % = - 4~rt (x - xo) 2 + (y - 3'0) 2 - v) (x - xo) z + (y - yo) z (B3)

- [ (x - x~ ] P x xo I - v + 2(i + V ) ( x _

cry>. = - 47rt (x - Xo) -~ + (3' - )'002 xo) --'-z + ("~ - 3'0) 2 "

State of stress at a point (x, y) due to four forces acting at points ( - x o , • as shown in Fig. 4 is obtained from relations (B33. The expression for ~y(x, 0) is

cry(x, 0 ) : P ( l + v ) [ 3 + v f Yo Yo } 2" [ ( x - x ~ (-'x+x~ ;'~] 2-~---T I _ I - - ~ t ( x - xo) = + )',~ + (x + s o y + ).8 - >o ~[((x - Xo) 2 + yo~) 2 + ((x + Xo)'- + yS):J_[ "

(B4)

494 R. C t t A N D R A et al.

Disp lacements a re obta ined as fol lows. Using s t r e s s - s t r a i n relation for the isot ropic shee t

(I + v)(3 - v) P s i n 0

4 ~ E t r

(I + v ) 2 P s i n O E 0 = - -

4~rEt r

1 - v 2 P cos 0

2 ~ E t r

(B5)

Using the s t ra in-d isp lacement relat ions

( 3 - v)(I + v) u(r, O) P sin 0 In r + A cos 0 + B sin 0

4 ~ E t

(I + v) 2 (3 - v)( l + v) t<r, 0) = - - P c o s 0 c o s 0 1 n r - A s i n 0 + B c o s 0 + Cr

47rEt 47rEt

(B6)

where A, B and C are arbi t rary cons tan t s . D i sp l acemen t s in x t y t coord ina tes are

(1 + v)ZP . u ( x h y D - - s m 0 c o s 0 + A - Cr sin O

4~rEt

(3 - v)(I + V) P i n r (I + v ) Z P c o s 2 0 + B + C r c o s O . t~xl, Yt) 4~rEt 47rEt

(B7)

Disp lacements in xy coord ina tes are

(I + v)2P (x - .ro)(y - 5"o) u(x. y) = 4 ~ E t (x - xo) 2 + (y - yo)"

(3 - v)(l + v) P In [(x - xo) 2 v(x , 5') 8wEt

+ A - C ( y - 5 ' o )

(x - Xo) 2

[(x - xo) 2 + (5" - 5"o) 2]

+ (5" - 5'0) 2]

+ B + C ( x - X o ) .

(I + v ) :P

4-~Et (B8)

The d i sp l acemen t s u and v due to a pair of point forces [P at (xo, Yo) and - P at (xo, -Yo)] are

(I + v)2P [ ( x - - Xo)(Y_-- 3__'o) , (x - xo)(y + Yo) ] u - 47rE'--------~ i x - - xo) 2 + (y - Yo)" - (x - Xo) z + ( y + yo)ZJ + 2A - 2C5'

(3 - v ) ( I + v) p In [ ( x - xo ) 2 + (5" - )'0) 2 ] (I + v)ZP v - 8~rEt (x XO) 2 + (3' + 5'0)2J 47rEt

{ (x - xo)" ( x - x o ) 2 ) x ( x - xo) - ' '2 -+ ( y - yo) 2 - ( x - xo) 2 + (5' + 5"o) 2 + 2B + 2 C ( x - xo).

(B9)

Enforc ing the b o u n d a r y condi t ions u(O, 0) = 0, v(x, 0) = 0, A, B, and C are found to be zero. Due to a pai r of forces [P at ( - x o , yo) and - P at ( - x o , - y o ) ] u and v are obta ined by subst i tu t ing xo by - x o in

relat ion (B9). Due to two pairs o f forces [P at (xo, Yo), - P at (xo, - yo), P at ( - xo, Yo) and - P at ( - xo, - Yo)] u and v are ob ta ined as follows:

4,'rEt ct et~ + et_~ a3 + ct., ct

v 8~Et L(aT-~'~ + ct4)(ct] + ~ - ~ / (Bl0)

"~Et (ct~ + ct~)(a~ + ct~) + (a] + a~)(a_~ + a4

where

t l I = X - - X O , O[2 = Y - - 5"O

~3 = x + Xo, ~4 = 5' + Yo.

v at x = Xo is obta ined f rom the above relation as follows:

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Analytical estimation of stress intensity factors in patched cracked plates