Indian Journal of Engineering & Materials SciencesVol. 2, April 1995, pp. 62-79

Stress analysis of an infinite plate containing two unequal collinear ellipticalholes under in-plane stresses at infinity t

V G Ukadgaonker, R R Avargerimath & S D Koranne

Department of Mechanical Engineering, Indianlnstitute of Technology, Powai, Bombay 400 076, India

Received 30 December 1993, accepted 23 August 1994

A closed form analytical solution ~o the problem of an infinite plate containing collinear unequalelliptical holes subjected to in-plane loadings at an angle {3with respect to x or y-axis on infiniteboundary of the plate is presented. The problem is formulated in the complex plane using the Kolo­soff-Muskhellshvili's complex stress functions and further the Schwarz's Alternating Method is usedto solve the problem of doubly connected region. The stress concentration factor for holes andstress intensity factors at all crack tips for varying sizes and centre to centre distances are evaluated.Some displacement formulation and the checked by Finite Element Method using displacement for­mulation and the two solutions are in good agreement The present results are compared with re­ported ones obtained by other methods. An analytical method for locating point in the vicinity ofellipses where the local and global strain energy density are equal is also presented.

I

lbl

y

x

x

x

p

The fracture process of the material is closely as­sociated with the interactio~ growth and propaga­tion of defects such as cracks, voids and inclu­sions. The study of interaction of above defectsexisting in a body subjected to a given form ofloading is therefore of particular importance. Inthis paper, closed form solutions to the case oftwo collinear unequal elliptical holes subjected to(i) uniform tension, P, parallel to collinear majoraxes (ii) uniform tension, P, perpendicular to ma­jQr axes, and (iii) uniform shear, T, on infiniteboundary of the plate are presented, which cangive solutions for cracks and circular holes as spe­cial cases.

The lengths of major axes of two holes are 2aand 2g, the lengths of minor axes are 2b and 2~respectively. The two holes are collinear alongmajor axes with centre distance as d as shown inFig.!. .••.

The boundary value problem in two dimension­al elasticity can be reduced to the solution of twocomplex stress functions given by Kolosoff-Mus­khelishvili method as given by Muskhelishvili 1.

The problem of infinite plate with one hole of anycontour can be solved by making use of a suitablemapping function which will map the infinite re­gion to outside of a circle of unit radius. Thisprocedure, however, becomes rather cumbersomefor multiply connected regions. Use of Schwarz's

tPart of this paper was presented at Canadian Congress ofApplied Mechanics, Ottawa, 1989.

Fig. I-Two elliptical holes in an infinite plate under (a) un­iaxial tension in Y-direction, (b) uniaxial tension in X-direc­

tion, and (c) uniform shear (T) at infinity

UKADGAONKER et al.: STRESS ANALYSIS OF AN INFINITE PlATE 63

First ApproximationFirst approximation. to the solution of an infi­

nite plate with two elliptical holes is obtained bysolving two problems of each hole in infinite plateand superposing each of these solutions on theother. The stress functions ;(~) and ..p(~) valid forsingle hole for each type of loading are obtainedby mapping the elliptical hole in Z-plane to the re­gion outside of a unit circle in complex ~-planeusing the mapping functions,

Alternatmg Method reduces a problem of mulnplyconnected region to a sequence of problems in asimply connected domains as given by Sokolni­koff2.

Many problems such as, interaction effect oftwo arbitrarily oriented and elliptical holes bycracks, stress analysis of a plate containing two el­liptical holes subjected to uniforms pressures andtangential stresses on hole boundaries, two un­equal circular holes with uniform pressures andtangential stresses, have been solved using Altern­ate Technique3'9• A novel method using sametechnique is also developed by Ukadgaonker andAwasarelO-14 for isotropic plate with circular hole,elliptical hole, equilateral triangular hole, rectangu­lar hole with rounded comers and orthotropicplate with elliptical hole. In this paper, a specificcase when the elliptical holes are collinear hasbeen studied in details.

t is boundary value of ~ on the circle (y) of unitradius in the mapped plane which defines theboundary condition foe t) as

... (8)

... (7)

... (6)

(00)

B'= uy - u~oo)2

C= .(,00)xy

(r)=PR[1- (2=+m));1 0,,1 4 0,,1± ~1

PR [ 1 ~1'(1+ mZ)..p~(~1)=4 =+2~1-~I + (m-~i)

± (1 + m~p(2 =+ m))~d~l-m)

circle; the constants B,C,B' and C' represent thestresses and rigid body rotation at infinity and aregiven by

d,aJ) + d,aJ)B= x y.

4 '

2G (aJ)C=--w .l+K '

l(t) = ;O(t) + w(t) ;O'(t)+ ..p0(t)w'(t)

The stress functions ;(~) and·..p(~) so obtained asfirst approximation for each hole and each type ofloading are as follows:(i) For uniaxial tensions (P):

and analytic stress functions ; O(~) and ..p0(~)aregiven by

° 1 l(t); (~)= --.f-( r)dt ... (4)2my t-."

..p0(~)=_.l....f let) dt_~(l;m~Z) ;o(~) '... (5)2m y (t- ~) (~ - m)

... (1)

a-bm= a+b

n=g-hg+h

ZI=Wl(~I)=R(1- +m). R_a+b0,,1 1-' --'0,,1 2 .,

and

Zz= wz(~z)=s(~~+.!!:.); S=g+h.~z 2 '

where, top row of signs correspond to uniformtension P, parallel to major axis and bottom rowof signs correspond to uniform tension, P, per­pendicular to major axis.

where,~= p ei6 in complex ~-plane represented by~= ; + iT}, suffix 1 and 2 correspond to first andsecond hole respectively, and general complexstress functions

F, + iF. '

;.(~)= - ( . ")log ~+(B+iC)R~+;o(~)2.7f 1+ K

... (2)K(F, + iF. )

..p(~)= - (I; ") log ~+(B'+iC)R~+ ..p0(~)2.7f 1+ K... (3)

where FI; and F" are the resultant forces in ; and 1/

directions, respectively on the boundary of unit

;Z(~Z)=PS[~ ±(2=+n))4 Z ~z

..pz(~z) = PS [=+ 2~z _~+ ~z(l + nZ)4 ~z (n- ~~)

± (1+ n~~)(2 =+ n))~z (~~ .•...n)

... (9)

... (10)

64 INDIAN J. ENG. MATER. SCI., APRIL 1995

... (19)

· .. (11)

· .. (12)

... (13)

· .. (14)

(~l - Cz)(1 + nZ)+ .(n-(~l - Cz)

± (1+ n( ~l - Cz)z)(2 + n)z Cz(~l - CZ)((~l - Cz) - n)

± Cz(2+n~)(~l- Cz)

These stress functions

through,

[tf/(~)]Or+ 08=4 Re w'(~)

08- Or+2i 'rr8= 2eZi8 [/.~\3 (w'(~);"(~)

- w"(~) tf/( ~)) + tp'( ~)]w'(~)... (16)

... (20)

... (23)

. .. (22)

... (21)

C1= dIR+J(dIR)z- 4m2

(ii) For uniform shear (T):

... (18)

a is half crack length.

f(~)is derivative of ;( ~)

But the simple superposition does not take intoaccount the interaction effect of the second hole inthe vicinity. For this purpose Schwarz's Alternat­ing technique is used to get the solution as secondapproximation.

and if the crack is present then Mode-I and

Mode-IT stress intensity factors can be obtained as C1 (2 + m))

~ + Z

1'l (~Z+Cl)K.- iKu=·2 -x f(~) ... (17)

a .•where subscripts 12 means translation from 1 to 2and 21 means from 2 to 1 and C. and C2 are dis­tances between centres of the two holes in the

mapped plane defined as

Second ApproximationStarting from the single hole solutions valid

near second hole, the stress functions ;2 (~2) andtp2 (~2) are translated to the centre of first hole byputting ~2 = ~. - C2 in Eqs (9), (10), (13) and (14).Similarly, the stress functions valid near first holeare translated to the centre of second hole by sub­stituting ~. = '~2 + C1 in Eqs (7), (8), (11), and (12).The resulting translated stress functions for eachhole are as follows:

UKADGAONKER et al: STRESS ANALYSIS OF AN INFINITE PlATE 65

.. ; (24)

(2=t=n)(1-2t1C2) (1+,*2)

± C2(1- tt C2)2 - 2{fh- C2)(1 + tt (fh- C2))

These translated ..stress functions give a non-zeroboundary condition of 121(11)on the first hole and/12(12)on the second hole which does not satisfythe stress free boundary conditions It (11)= 0 andh(t2)=0. In order to correct these boundary con­ditions, new problems of an infinite plate with firsthole or second hole and with the boundary condi­tion which is negative of 121(t1) or 112(t2) respect­ively applied to them are solved using theCauchy's integral equation and the correctedstress functions are obtairied. The actual stressfunctions valid near each hole are then obtainedas the sunt of· translated' streSs functions artd cor­rected· fwttmons. Thus the aetUal streSs functions,d tD 8l1d "'1(t1)valid fot first hate, '2 (~i) artd'/12('t~)valid fotseetilid hole are as givelibelow:"-

(i) For utliaDal tel1sidrt(P): .

psi (2+n) m,dt1)-4" (tl-C2)±(t1-C2) t1

(2 ~ n)(2C2 - mC1t1 -' 3C~t1 - m2C~tt+ mt1 + 2m2C2)

(1- d2 t1)2(C~- m)2

... (28)

(2 =Fm)( - 2Ct - n2cit2 + nt2 - nCit2- 3cit2 - 2n2td

(1+ t2 Ct)2( ci - ,,)2

±~_ 1=F (2=Fm)t2 Cd1 + t2 Ct) mCd1 + t2 Ct)

(1+ m2)

2([m+ Ct)(1 + t2([m+ Cd)

(2 =F m)(1 +m2)±---------2m(1 + t2([m+ C1))([m+ Ct)

=F(2 =Fm)( 1+ 2t2 Ct)]C1 (1 + t2 Ct)2

PR! . 11/J2(t2) =4 =F 2(tz + Ct)-

+ (t1 + C1)(1 + m1)(m-(t2 + ctf)

± (1 + m( t2 + ~1)2)(~=Fm)(t2+C1)((tZ+Ct) -m)

+ td 1+ ~2) ± __ (m_2_+_1_)(_2_=F_n)(m- t1) 2([m+ tt)(Jm+ C2)2

=F (m2+1)(2=t=n) _ (1+n2)

2([m- tt)([m- C2)2 2(Jn+ C2- td..

± (1+n2)(2=Fn) ]_td1+mti);lt(tt)2n([ir+ C2':'" tt) (ti-- m)

... (27)

where, ;11(t1) is the' derivative of COl'rectedstressfunction ;1l( t1)' .

PR I (2=F m) n;2(t2)=4 (t2+ Ct)±(t2+ Ct)- t2

'" (26)

'" (25)

(1+n2)(2=Fn) ]± 2n(;;,_ C2)[1 + tt ([ir- C2)]

psi 1'/11(t1)'" 4" =t= 2ft1 - C2)-

+ (t1- C2)(1 + n2)=F (1 +'n(t1- C2)2)(2=Fn)(n-{t1- C2)2) (t1-C2)[n-(t1- C2)2]

+Ct=FCd2=Fm} ~± (2=Fm)(t2+Ct) t2 Ct(1+Cttz)

tz(1+n2) (nZ+1)(2=Fm)+ z ±------

(n- t1)' 2(Jn+ tz)([ir- Ct)1

66 INDIAN J, ENG. MATER. seI., APRIL 1995

(ii) For shear (T ) on infinite plate boundary:

;dl;l} = iTS[_1_

(- 2CI- nCi l;2- 3ci l;2

;2 (l;2) = iTR [ 1 + - n4Ci l;2+ nl;2 - 2n2 CI)(l;2 + CI) (1 + l;2 CI )2( ci - n)2

+.!,+ . 1l;2 mCI (1 + l;2 CI).

(1 + n2)

2n([;i- C2)(I'+l;2 ([;,- C2))

(1- 2C2l;1) ). - _ •~ .~ .", ... (30)

1/I1(l;1)= iTS[(l;I- C2)+ [1+ n(l;I- C2)2]

+ . C2 + 1(l;1 - C2)2 C2 (1 ~ l;1 C2)

... (3~)

(1 + m2)

2m([m+ C1)[1 + l;2([m+ C])]

(1 +2CI l;2) )+ C1 (1 + CI l;2f

[ [1 + m( l;2 + CI)2]'ljJ2(l;2) = iTR (l;2 + CI)+ (l;2+ CI)[(l;2+ clf - m]

CI 1

(l;~ +cli- ~ (1 + l;2CI)

(2e2iP+n)e-4il/ (3+mC~)C2

=F C~ - (C~ - m)(1'- C2l;1)

2(m+ m2C~) C2+ 2 2

(C 2 - m) (1 - l;1 C2)

'. 2' . 2_ (l + n )(nl;2 - 2nC) + l;2C I)

(n-l;~)(Ci-nf

" (1 + m2) )- 2m( l;2 - ([m-~I))

l;2(1+ nl;~) .'

- (l;r-n) ;22 (l;2) ... (33)

, For ,the case of an infinite plate C9ntainingtwounequal elliptical holes, subjected to ~axial ten­sion acting at an angle p with respect to X- orY-axis following facts can be concluded.

(i) an infinite plate containing two unequal el­liptical holes subjected to uniaxial tensionacting at an angle p with respect to Y-axis isequivalent to the case of an infinite platecontaining two unequal elliptical holeswhose major axis is inclined at an angle pwith respect to Y-axis and plate being sub­jected to uniaxial loading along Y-axis.

(ii) An infinite plate containing two unequal el­liptical holes subjected to uniaxial tensionacting at an angle p with respect to X-axis isequivalent to the case of an infinite platecontaining two unequal 'elliptical holeswhose major axis is inclined at an angle Pwith respect to X-axis and plate being sub­jected to uniaxial loading along X-axis.

Considering above two facts, stress function;1 (l;I) and ;2 (l;2) for the two holes are found asgiven Eqs (34) and (35).

;dl;l) = PS[(l;I- C2)±(2e-2iP=Fn)e4ip m4 (l;I- C2) l;1

... (31)

(1 + m2)

2([m- Cj-l;2)

... (29)

(n2+ 1)(2=Fm)=F--~--2([;,-l;2)([;,+ CI)2

± (1 + m2)(2=F m) )2m([m- CI -l;2)

_ l;2(1 + nl;~)..t.. (1-). (l;~- n) 7'22 0,,2

+

(2C2 - mc:il;1 - 3C~l;1 - m2C~l;1

+'ml;1 + 2m2 C2)

(1- C2l;1)2(C~- m)2

1 1+-------l;1 nC2 (1 -l; 1 C2)

(1 + m2)(ml;1 + 2mC2 + l;1 C~)

(m-l;i)(m- C~)2

(1+ n2) )+ 2n(([;,+ C2)-l;I)

l;1(1+ ml;i) ;11 (l;I)- (l;i - m)

and,

UKADGAONKER et a/;: S1RESS ANALYSIS OF AN INFINITE PlATE 67

2 2ifl 2 2ifl±~+ e·.

{;l nC2 (1 - {;l C2)

(1 + n2)(e2ifl=fn)=f--------n[C2-In.e2ifl] [1- {;l (C2-[;J.e -~ifl)]

Similarly fQr shear loading of an infinite platecontaining two unequal elliptical holes whose axesare inclin~ at an angle f3 we have

[ 6ifl -6ifl;d{;d= iT.S e + e

x( - 2C2- mC~{;l - 3Ci{;1 ~ m2Ci{;l

... (36)

2e -2ifl 2e -2ifl±--------{;2 mCl (1 + Cl {;2)

(1+ m2)(e -2ifl=f m)±---~--------m[( Cl- j;,..e2ifl)(1 + {;2(Cl - j;,..e -2ifl)

± (2e -2ifl=f n)e -4iflC~

(3+nC~)Cl

(n'- C~)(1+ {;2 Cl)

iTR [e -6i~ e6ifl

;2({;2)= (t2+Cl) +(1+ {;2Cl):

x(2Cl - nC1'{;2 - 3C~{;2 - n2C~{;2 + n{;2

e2i{l(1+m2e8ifl)+------------2m·e'4i~C,--Jn·e2iflJ[1 + {;2( C~- In·e2ifl)]

.. . (38)

... (39)

.... (37)

Using these actual stress functions as second ap­proximation to the stress field one can obtain thestresses ·up U(J and 'l"r(J for each hole. However, asthe hole is· stress free we can write the expressionfor stress concentration factor as the ratio of tan­

gential stress to applied stress given by

u(J= 4 Re [;'({;)]p 'w'({;)

... (35)

where each. parameter has its own meaning as ex­plained earlier .

In the above two equations and all the equa­tions which will be dealt .further the top row ofsigns corresponds to tension parallel to positivedirection of X-axis bottom row of signs corre­sponds to tension parallel to negative direction ofY-axis ..

68 INDIAN J. ENG. MATER. SCI., APRIL 1995

-$ $-

---$-~ --$-Fig. 2- Particular cases for each type of loading

Results and DiscussionThe solution presented here is for the case of

elliptical holes as a general case. However, by al­tering the hole parameters (a, b,g, and h) it is pos­sible to solve the problems such as of· circularholes where in the length of major axis is equal tolength minor axis (a= b, g= h), crack problems,length of minor axis relatively smaller than majoraxis (" l/1000th of major axis). A total of nineparticular cases for each type of loading are possi­ble by this solution as shown in Fig. 2. Generalclosed form solution for the cases of tension, P,are also obtained as given below· which will beuseful for design calculations.(i) Closed form solution for tension, P, perpendi-

cular to collinear major·axesFor first hole at 8= 0° (inner tip)

U(J S ( (2+n)P=R(1-m) 1+,. ~\2+m+(2+n)

((1- Cz)z[- mCi - 3ci- mZC~+ m]+

[2(2 - mCi - 3C~ - mZC~ + m+ 2mzCz]

x 2Cz(1- Cz)) .(l-Czt(C~- m)z

2 1 (2+n) (1+nz)+ + z z+----(1- Cz) n(1- Cz) 2[1 + j;,- Cz]z

+ (1+nz)(2+n) +2Cz(2+n)) ... (40)2n[1 + j;,~Cz]z (1- CZ)3

For second hole at 8-180° (inner tip)

U(J R ( (2+m)P = S(1- n) 1+ (Cl-1)Z~ n+(2+ m)•

[(-nzCi+ n-nC~- 3Ci)(1- C1)z

- (- 2C1+ nZCi - n+ nC1 + 3ci - 2nZC1tx 2C1(1- C1)]

(Ci - n)Z(f= c1t

2 1 (2+ m) (1 + mZ)+ + Z . z+-----(1- C1) m(1- C1) 2(1- J;r- C1)z

+ (2+m)(1+mz) +2C.(2+m) ... (41)2m(1- j;,- C1)z (1- C1)3

(ii) Closed form solution for tension, P, parallel tocollinear major axes

For first hole at 8= 90°

U(J [_S_' ( _(2-_n)_p"=Re R(1+m) 1-(i-Cz)Z m-(2-n)

[(1- iCz)z[- mC~- 3C~- mZC~+ m]­

[2Cz - imCi - 3iC~ - imzC~+ im+ 2mzCz]

x t - 2Cz (1- iCz))](1- iCZ)4( C~- m)z

2 1 (2-n) (1+nz)+ + z+ z+------(1- iCz) n(1- iCz) 2[1 + iCJ;;- Cz)f

UKADGAONKER et aL: STRESS ANALYSIS OF AN INFINITE PLATE 69

U6= Re [ R (1(2- m)P 5(1+n) -I' .•.•\2 n-(2-m)

_ (1 + n2)(2 - n) 2iC2 (2 - n))!2n(1 + i (fi,- C2})2 (1- tC2)3

For second hole at 8= 90°

... (42)(2e -2iP=Fm)e4iP (3 + nCD cf

± 2- 2 2C I (n- C 1)(1 + ~2CI)

2(n+ n2C~)Cf 2(1- nC~)Cf

(1 + ~2CI)2(n- Cf) - (n- Cf)(l + ~2CI)

... (45)

Similarly for shear loading it is observed that fortirsthole

[(1+Cli)2( - n2C~ +,,-nC1 +nCi - 3C~)

- (- 2CI- in2Cf + in- inC1- 3iCf

- 2n2Cd2CI (1+ tCI)]'

(C~- n)2(1+iclt

(1+m2). (2-m)(1+m2)

+ 2(1 + i J;,+ CI))2 .. 2m(1 + i([m+ CI))2

+2iCd2-m))!(1+ tCI)3

From Eq. (34) for first hole for uniaxial loading

U6 [ 5 [(2C-2ift=F n)liP m-p=Re R(1-m/~f) 1=F (~I"-C2)2 + ~

e -6ifJ( - mC4 - 3C2 - m2C2 + m)+ 2. 2 _ 2

(C1-m)2(1-~IC2)2

e -6iP( - 2C2 - mC;~1 -3C~~1 - m2C~{;1

+ m~1+ 2mC2)( - 2C2)

(C~ - mf(1- ~I C2)3

1 e -2iP

- ~f - noe-4iP(1_ ~1 (2)2

2e2iP 2Ce2iP 2 Ce2iPo~2=F-2+-2-+---:-~I ~I n(1- ~IC2)

(1 + n2)(q2iP=F n)=F---~---n(1- ~1 (C2 - fi,oe2iP)f

=F (1 + n2)(e2iP=Fn) =F(2e2ifJ=r n)e -4ipn[1- ~1 (~2 + fi,oe2IP)f C2

e-2iP(1+ n20e-SiP)

2noe -4i~1_ ~1 ( - C2- fi,oe -2iP)f

e -2iP(1+ n20e-SiP)

2noe-4i~1 +~I (- C2+ fi,oe -2iP)f

2 2(1- 2 ~1 C2)]!+ (1- ~IC2}2 (1- ~IC2?

... (44) and for secon<l hole it is observed that... (46

and for second hole

~I= Re [ R,. 2 [1 =F(2e2iP=Fm)e -4iP+.!!.P 2 5(1- n =F/~2) (~2+ C1)2 ~~

70 INDIAN J. ENG. MAlER sa., APRIL 1995

1 e2ill

- ~2+ n·e4ill(1 + ~2CI)2

Numerical results

Following section gives the numerical resultsand discussion for the particular cases.

Two equal and unequal circular holes in an infi­nite plate-Stress concentration - factors for twoequal and unequal circular holes in an infiniteplate subjected to tension parallel to and perpend­icular to line of holes and shear at infinity are ob­tained by taking the hole parameters as a = b andg= h. Tables 1-3 and Fig. 3 gives the results ob­tained for two equal circular holes problem forvarying centre to centre distance. From these it isseen that for the case of tension in X-direction,the SCF is lesser than single hole solution at closedistances, whereas for tension in Y-direction theSCF is higher and converges to single hole solu­tion as the distance between the holes increases.For shear (T) at infinity the 08/ T values increasesup to e= 2 and then decreases. and converges tosingle hole solution. These solutions of second ap­proximation. to stress field using Schwarz's alter­nating technique are in good agreement with thoseobtained by Nemat-Nasser and HorilS making useof Pseudotraction method. Ling16 using bipolarcoordinates and Ukadgaonkerl7-19using complexvariables, up to about 3% difference at closer dis­tance and 0.5% for larger distance between holes.

Fig. 4 gives the variation of SCF with distancebetween holes (e) for two unequal circular holes.From this it is evident that presence of a smallerhole near a larger hole results in increase of stresslevel on larger hole results in X-direction, higherstress level on smaller hole for tension in Y-direc­tion at closer distance between holes. For. shear atinfinitythe SCF is higher on smaller hole.

Crack approaching circular hole-Presence of acrack near a circular hole is accounted for byputting a= b and h ~ 0.001 x g. For the case of ten­sion parallel to crack, the presence of crack near acircular hole will result in reduction of SCF on thehole and increase of SIP (K1) value for crack itselfand converges to single hole and single crack solu­tion as the crack moves away from the hole. Forthe case of tension perpendicular to the crack, thepresence of crack near circular hole at tloser dis­tance increases the SCF on hole and SIP for thecrack and the two values decreases to single holeand single crack solution at larger distance be­tween the hole and crack. The K1 values at thetwo crack tips are compared with those obtainedby Isida20 in Table 4. The results are in goodagreement at larger distances.

For the case of shear the interaction effect in­creases SCF value on the hole and SIP (Kn) valueat the crack tips.

Circular hole and elliptical hole- The ~tressconcentration factors for a circular hole. along themajor axis of an elliptical hole in an infinite plate

... (47)

e2ill(1+ m2e8ill)

2me4ill[1-+ ~2(CI + ji,·e2ill)f

+ 2(1 +2~2~1)1](1 + ~2CI) J

e2ill(J+ m2e8ill) .

2me4ill[1+ ~2(CI - j;,,·e2ill)f

Hence

Re[ S

x [1'1'(20~'"''f.)e'~~+ (-2+20)

(2e2ill~n)e -4ill(-(3 + mC~)C

+ m+ 2e2ill+ + 2(m+ m2C~)+ 2(1 + mC~)CDCD2

e6ill(- 2CI - nC1~2- 3ci ~2- n2Ci ~2+ n~2- 2nCI)'2CI

(Ci - n)2(1+ ~2CI)3

where,

A=(~I- C2)2,B= ~i.C=(C~- m),D=(1- ~I C2),E= [1- ~I (C2- ji,·e -2ill)f

and

F= [1- ~I(C2 + ji,'e -2illf

Now ~1=pei8. Solving the above equation for ~1'the location of point for Strain Energy Densitycan be found in the vicinity of first"ellipse, whenan infinite plate containing them is subj~ted touniaxial tension along x- or y-axis. Similarly forobtaining location in the vicinity of second ellipse~1 is substituted by ~2' C2 by. - CI, P by - p, nby mand mby n, Rby Sand SbyR, inEq.(48).

On similar lines one can proceed for the case ofshear loading.

UKADGAONKER et al.: STRESS ANALYSIS OF ~N INFINITE PlATE 71

FIRST 1tOI.£:

•• h2SlCONDHCU

•• he2

Table I-Stress concentration factor for two equal circular holes subjected to uniform tension along the line of holes by comeplex variable method

~~*EBy present method

For first hole

~,

For second hole

e

0'1o,/P. 82o,jP

0.5

9702.50752~8302.507521.0

9702.522378302.522372.0

9602.564,228402.564223.0

9502.614668502.614664:0

9402.664808602.664805.0

9302.710158702.710156.0

9202.749088802.749087.0

9202.782118802.782118.0

9102.809478go2.809479.0

9102.832748go2.$327410.0

9102.8.52098902.8520915.0

9002.922419002.9224120.0

9002.941959002.9419525.0

9002.958199002.9581930.0

9002.99589002.995850.0

9002.99769002.9976

Single hole

9003.0009003.000

solution Nemat­

Nasser

2.5500

2.6088

2.6500

2.8272

2.947$

3.??oo

Ukadgaonker9

o/Pat90°

2.658

2.787

2.907

2.968

2.978

3.0000

Table 2-Stress concentration factor for two equal circular holes subjected to uniform tension perpendicular to line of holes bycompleX variable approach

,.FIRST HOlE:

•• b.2SEmND HOlE:,-

- ~, •• h.2

1 JJ p r~j

Present solution Nemat-e

o/Pat (fNasserUkadgaonkerB Ling"Haddon I

0.5

4.57689 '4.4227

13.621783.92113.9973.8693.264

2

~3.103133.26413.4813.0663.066

3

3.00262 3.020

4

2.98243 2.9853.020

5

2.98093 3.004

6

2.98378

7

2.98726

8

2.990382.9922 3.001

9

2.99296

10

2.995U5 3.0453.003

15

3.00071

20

3.000102.9981

Single hole

3.00003.00003.0003;0003.0000solution

72 INDIAN J. ENG. MATER SCI.,APRIL 1995

Table 3-Stress concentration factoi' for two equal clrc:ular holes subjected to Wuform shear at infinity by coIIlplex variable··method

By present method

First holeSecond hole

e

61 de!.a.,/T62 deg.a,jT0.5

34-4.i5018.1464.250181

32-4.537841484.537842

36-4.622681444.622683

38-4.546021424.546024

40-4.459101404.459105

42-4.384391384.384396

42-4.321851384.321857

44-4.271691364.271698

44-4.232961364.232969

44-4.200921364.2009210

44-4.174461364.1744615

45-4.084731354.0847320

45-4.04&2213S4.0482225

4S-4.0288813S4.0288830

4S-4.017491354.0174950

45-4.006121354.00612

Singlehole

45-4.000 1354.??oosolution

y

5

-4.1933-4.2029-4.2126-4.1936-4.1746-4.1559-4.1373-4.0866-4.0487

-4.000

Unlaxial t.".ICIftparall.1 to X-axl.

,

Zo , I 110 IZ 14 II

DlRallC. b.twe." ·hol•• Ie)20 nx

Fig. 3-Variation of alP and alT with e for equal circular holes

UKADGAONKER et aL: STRESS ANALYSIS OF AN INFINITE PlATE 73

y

5

Go..•~

-- --------------

--- For smaller hole-- --- For larger hole

Uniaxial tensionparallel to V- allis

------------------------------

Uniallial tensionparallel to X- axis

22 x201842 6 8 10 12 14 16

Distance betw.enholes leI

Fig. 4-Variation of o,jP and o,jT with e for unequal circular holes

2o

For crack For crack

e

0.5

12

345

c 6

7

89

10

20

30

50

For circular

hole o,jPat90·

2.7842.8182.8672.8992.9202.9362.9462.9552.9612.9662.9702.9872.9912.995

K •.AlP

-2.724-1.175-0.252-0.037

0.0220.0380.0390.0370.0330.0290.0260.0090.0040.001

K.,BIP

-0.083-0.032

0.0290.0270.0250.0250.0230.0210.9190.0170.Q150.006

0.0030.000

For circularhole

o,jP3.4193.2673.1423.0913.0653.0493.0393.0323.0273.0243.0213.0103.0063.002

For crack

K •.AlP KII. HIP

6.836 2.8264.795 2.7433.384 2.6542.944 2.6082.763 2.5812.673 2.5642.623 2.5532.593 2.5442.573 2.5382.560 2.5332.549 2.5302.518 2.5152.512 2.5112.508 2.508

ISID~S Results

K •.AlP K., HIP

4.387 2.9573.384 2.8072.958 2.6822.757 2:6072.695 2.5822.594 2.5572.570 2.543

2.507 2.507

For circular

hole 091Pat 45·

-4.052-4.079-4.084-4.071-4.058-4.047-4.035-4.031-4.023-4.022-4.018-4.003-4.001-4.000

KII•AIT

-1.3461.2682.6292.8352.8312.7882.7442.7072.6762.6512.6312.5472.5272.513

KII. Pr1'

2.580.2.6492.6842.6772.6602.6412.6252.6112.5982.5882.5782.5362.5232.513

are obtained by the present solution and are givenin Table 5.' It may be concluded from these resultsthat the interaction effect of the two holes is to re­duce the stress level on each hole at closer dis­tances for tension, p,in X-direction. However, fortension perpendicular to major axis, the effect of

interaction is to increase the stress level on eachhole remarkably at closer distances and reduces tosinglehole solution at larger distance.

For the case of uniform shear the interaction ef­fect is about 9% on circular hole and 24% on el­liptical holes. The SCF on circular hole and ellipt-

74 INDIAN J. ENG. MATER SCI., APRIL 1995

Table 5~Stress concentration factor for a circular hole and a elliptical hole in an infinite plate

g~'.~'p - - P q. 2

- • • • g h. 1 J~lf:::T _

e

Circular hole Elliptical holeCircular hole

08/PatO•

Elliptical Circular holehole

08/Pat 180· 8( deg 08/T

Elliptical hole

0.5

12

34

5

67

89

10

20

30

50

9594

9392

92

91

91

91

91

90

9090

90

90

2.626

2.655

2.714

2:766

2.807

2.840

2.865

2.885

2.901

2.914

2.924

2.970

2.983

2.998

75

75

77

79

81

82

84

85

86

87

87

89

90

90

1.675

1.690

1.719

1.751

1.782

1.810

1.835

1.856

1.874

1.889

1.902

1.965

1.982

1.993

3.619

3.276

3.080

3.033

3.019

3.013

3.011

3.009

3.009

3.008

3.008

3.006

3.005

3.002

9.397

7.140

- 5.701

5.306

5.161

5.108

5.065

5.047

5.035

5.027

- 5.023

5.006

5.003

5.001

37

38

41

42

43

44

44

44

44

45

45

4545

45

-4.295

-4.354

-4.337

-4.282

-4.231

-4.189

-4.157

-4.131

-4.111

-4.094

-4.081

-4.027

-4.013

-4.000

150

155

157

156

155

155

154

154

154

154

154

153

153

153

4.604

5.177

5.550

5.421

5.243

5.094

4.980

4.895

4.828

4.776

4.735

4.572

4.534

4.502

Table 6-Stress concentration factor for two equal elliptical holes by complex variable method

a-b-2 g-h-l

l~~T _

For hole 1 For hole 2 For hole 1 For hole 2 For hole 1 For hole 2

o/PatO· o/Pat 180·

6.682 6.682

5.894 5.894

5.358 5.358

5.189 5.189

5.118 5.118

5.082 5.082

5.060 5.060

5.047 5.047

5.037 5.037

5.030 5.030

5.025 5.025

5.008 5.008

5.003 5.003

5.001 5.001

e

0.5

12

34

5

6789

10

20

30

50

Single holesolution

81

degree.oJo

102

100

9896

94

9492

92

92

92

,:?2

90

90

90

90

1.757

1.776

1.813

1.846

1.873

1.894

1.911

1.925

1.936

1.944

1.951

1.983

1.991

1.997

2.000

82

degree

78

80

82

84

86

86

88

88

""88

88

88

90

90

90

90

1.757

1.776

1.813

1.846

1.873

1.894

1.911

1.925

1.936

1.944

1:951

1.983

1.991

1.997

2.000 5.000 5.000

81

degree

26

24

26

26

26

26

~6

26

26

26

26

26

26

26

26.56

-4.960

-5.120

-5.065

-4.945

-4.846

-4.772

-4.718

-4.677

-4.647

~4.623

-4.605

-4.532

-4.515

-4.505

-4.5

82

degree

154

156

154

154

154

154

154

154

154

154

154

154

154

154

153.43

4.960

5.120

5.065

4.945

4.846

4.772

4.718

4.677

4.647

4.623

4.605

4.532

4.515

4.505

4.5

UKADGAONKER et al: STRESS ANALYSIS OF AN INFINITE PLATE 75

y

II \ Te"alOft perpendicular, to major axia

•.•.a6

••

I0' . 0 000., '-- Shear ( T)~ ~

,"~~;~L

3

" ~

21-

r Tuaion parGUelto major axis

Io

e

Fig. 5-Variation of 0,/ P and 0,/ r with e for equal elliptical holes

y

II~~ ,~

••IIX 3L"~

-----------

--- For ama"er hole----- for I arger hole

Tenaion perpendicularto major axla

-.....g.------~-~--a81T For shear (T)

TenalOll paraUe'to major axis

---------------------------------

,o 2 , I • 10 12 14 II II 20

C

Fig. 6- Variation of 0,/ P and o,/r with e for unequal elliptical holes

22 X

ical hole increases at closer distance and then re~

duces to single hole solution for larger distances.Two equal and unequal elliptical holes-For two

equal elliptical holes with their major axes colli­near along X-axis, the SCF defined as the J~tio ofmaximum tangential stress on the hole boundary

to the applied stress at infinity for the cases of un­iaxial tensions and shear (T) are tabulated inTable 6. Figs 5 and 6 give variation of olP and01T for varying distance between the. holes.When the holes are close to each other for tension

parJ1llel 'to major axes the effect of interaction of

76 INDIAN J. ENG. MATER. sel., APRIL 1995

Scalo'Dlmonolono 1·5 s 1

Tangontlal }.Sir,. dlotrlhllon 2.1

Fig. 7 -Tangential stress distribution on hole boundary as aratio of tangential stress to applied stress'

Fig. 8-Tangential stress distribution on hole boundary as theratio of tangenti8I stress to applied stress for unequal holes

(with tension in X-direction)

•

each hole is to reduce the level of stress on theother hole and as the ~o holes move away fromeach other the effect reduces, the stress level oneach hole increases and converges to single holesolution. However, the effect of interaction for thecase of tension in Y-direction is vice-versa, i.e., atcloser distances the stress level on each hole ishigher and reduces to single hole solution at largerdistance.

For the case of two unequal elliptical holes,when the second,.hole is at larger distance fromthe first hole, the stress field near first hole shouldgive a single hole solution and should not be af­fected by the variation of size of second hole.However, with the existing stress functions this ex­pectation is not satisfied, so as .1 measure of cor­rection, the quantities S and Rin ~1 (~1) and rh ( {;2)'

are replaced by Rand S, respectively. The results

UKADGAONKER et aL: STRESS ANALYSIS OF AN INFINITE PLATE

p

77

.,,--:------.•..,---t---

6.0

1.0

g.4

p

4S

10"

Fig. 9-Tangential stress distribution on the hole boundary oftwo unequal elliptical holes subjected to tension perpendicu­

lar to collinear major axes

j

J

T

r

r

rT _~ _ -~------ ---------

Fig. to-Tangential stress distribution on the boundary oftwo elliptical holes subjected to uniform shear (T) at infinity

are found to be satisfactory with this change.Moreover, this correction does not affect theequal holes solutions as R is equal to S in suchcases.

By the present solution alP (maximum) is ob­tained for equal ~d unequal holes all around thehole boundary for all the three types of loading asshown in Figs 7-10.

Two collinear cracks~ The solution to the prob­lem of two collinear cracks in 'an infinite plate is

obtained by putting b= h=O. The Mode-l stressintensity factor as a ratio of KI to P j;O. at all thefour tips for tension, p, perpendicular to crack lineand Mode-ll stress intensity factor as a ratio of Knto T ~ for uniform shear (T ) at infinity for 2a/dvarying from 0.9 to 0.1 obtained by the presentsolution are listed in Table 7. Figs 11 and 12 givethe variation of KIf P ~ for varying e and 2a/d,respectively. For the case of tension perpendicularto crack line, at very close distance between the

78 INDIAN J. ENG. MATER seI., APRIL 1995

Table 7 -Stress intensity factors! at the four tips of two equal collinear cracks by complex variable approach

KitPIifci"

ll~'1~

P----20ld

By present methodNemat-Nasser2ISIDNBy present method--Inside OutsideInsideOutsideInsideOutsideInsideOutside

0.9

1.2461.073--1.41.120;9861.016

0.8

1.1541.0561.22891.08111.211.071.0141.017

0.7

·1.0971.0421.1331.05791.1151.051.0221.016

0.6

1.0601.0301.08041.04091.071.041.0201.013

0.5

1.0361.0211.04801.02801.051.031.0161.011

0.4

1.0211.0141.02721.01791.0251.021.0111.007

0.3

1.0111.0081.01381.01021.021.0151.0061.005

0.2

1.00431.0041.00571.00461.0051.0041.00271.002

0.1

1.0001.0001.00131.00121.01.01.0001.000

cracks, the interaction effect is up to 25% on theinner tips. and 7.5% on the outer tips. The effectreduces as the cracks move apart.

For' the case of uniform shear, at infinity theMode-II SIP increases initially upto 2a/d= 0.6 andthen decreases and converges to single crack solu­tion

Strain enel'lY deDSity

. For calculation of strain energy density the ap­proach followed here is as follows:

The emphasis here to 'locate the point in thevicinity where local and global strain energy densi­ties are equal as given·.by 5ih2l• Now, the stt:~energy density, local and global can be equal at apoint only when local stress and global stress atthat point is equal. If the location of point wherethe stress concentration factor is 1,. is found, thenthat point is having local and global strain energydensity equal.

Here, global stress is the stress applied at infi­nity and local stress is the stress obtained by usingthe stress functions obtained.

Hence, the task here is to find the point nearthe edge of ellipse or in region between the twoellipses where stress concentration factor is 1.

Consider the equation for stress concentrationfactor as given in Eq. (45).

Substituting a';p= 1 and rearranging the equa­tion by taking LCM we get a polynomial in ~l as ­given by Eq. (48).

B. A CN..~;2:"Jp

alb= 1______ alb ••2

H ,.J ,., H H ,., HX2.,.

4

6

'024681 2 3456Tip Diat .•

.".,Tip C

51- '...,,\,,

TiPA,C\. \ ,\

~3

Fig. 12- Variation of Kl P for two collinear cracks

Fig. 11- Variation of KI/ p!j;i;iJfor cracks

'4, •..

~~,•• I ---In'" Tip(A)------ Out .de Tip ( lQ

UKADGAONKER et al: STRESS ANALYSIS OF AN INFINITE PLATE 79

Conclusion

The present results proves that Schwarz's alter­nating technique converges jn second approxima­tion itself. The solution provides a close form so­lution unlike the other solution provides a closeform solution unlike the other solutions which inthe form of infinite series in which the converg­ence of the 'series in itself becomes a separateproblem.

Present solution is computationally very easy, asonce it is programmed it can give solutions tomany particular cases by just inputting the holeparameters.

Regarding 'strain energy density calculation, forlocation part in the vicinity of the ellipses wherelocal and global strain energy densities are equalEq. (48) was solved and few particular case of theunequal collinear ellippcal holes were considered.It was observed that the point lies near the biggerhole. Further, when the distance between the holesis increased, there is shift in the location of pointwhere local and global strain energy are equal.The shift continues till intc!raction between thetwo holes exists. When ipteractionbetween thetwo holes is negligible than the location of point.remainssame as in earlier case.

AcknowledgementThe financial aid provided by the Board of

Research in Nuclear Science, Department ofAtomic Energy, India to this work is gratefully ac­knowledged.

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matical theory of elasticity (P Noordhoff Ltd., Groninicen,Netherlands),1953.

2 Sokolnikoff I S, Mathematical theory of elasticity(McGraw Hill, New York), 1956.

3 Ukadgaonker V G & Naik A P, Int J Fract, 51 (1991)219.

4 Uki\dgaonker V G & Naik A P, Int J Fract, 51 (1~1)285.

5 Ukadgaonker V G & Koranne S D, Int J Fract, 51 (1991)R37.

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16 LingCB,JAppIPhys,19(1948)77.17 UkadgaonkerV G, 20th ISTAM Congress, BHU, 1976.18 Ukadgaonker V G, MED J Inst Eng (ln4ia), 69 (1988)

107. s.:

19 Ukadgaonker V G, Indian J Techno~ 20 (1?82) 254.20 Isida M, Mechanics of fracture-I (P ~oordhoff Int Pub,

Netherlands), 1973.21 Sili G C, Theory Appl Fract Mech, 4 (1985) 157.