Two Unequal Inclusions in Infinite Plate

An analytical so lution is presen ted for the stress distribution in an infini te plate which is pierced by two inclusions of ri g id circular discs of diffe r ent diame ters . The inclusions a re assumed to produce uniform normal pres sure along the circula r bound a ri es . The first fundamental boundary value problem is formul ated in complex pl ane in te r ms of complex stress func t i ons and t hen the Schwa r z ' s alte rnating method is u sed for obtaining the solution . The complex stress functions a re dete r mined for t he case of a single circular hole , neglecting the effect of the othe r hole . These leave s ome residual stresses at t he edge of the second hole , which are corrected b y assuming suitable addit ional stresses the re . The new stress functions a re computed , wh ich , i n general , do no t satisfy t he bounda ry condition at the edge of the fir st hole . These are corrected again and the procedure i s r epeated with the two holes a lternately . Thus the successive approximat ions to the st res s functi on s a r e ob tained. Numerical results of the stress field are compu ted upto the second approximati on by the use of computer. The stress concentration f actors for the tangen tial s tres s es at t he boundarie s of the holes a re ob tained by iii

Numerical results of the stress field are compu ted upto the second approximati on by the use of computer.
The stress concen tration f actors for the tangen tial s tres s es at t he bounda rie s of the holes a re ob tained by iii varying distances , interna l pressures and hole diameter ratios . The variation o f stress dist ribut ion alon g the radial direction are plotted f or dif f eren t a n g ul ar coordinates .
It is found t ha t the maximum st re ss concentration occu r s at a poi nt on the edge of the small er h ole n ea r est to the la r ger on e .  A general solution for square holes with rounded corners is due to Heller,Brock and Bart[20] which is an improvement on an earlier solution by Greenspan [21].
(b) Two Holes: The solution in terms of a biharmonic function for two equal circular holes in an infinite plate under biaxial and uniaxial stresses was first discussed by 3 Chih Bing Ling [22] in 1948 . He took advantage of the double symmetry and used bipolar coordinates , a method used earlier by Jeffery[8] and l atter corrected by Mindlin[9] for the case of single circular hole near the straight edge of a semiinfini te plate . These cases have also been considered by Us tino v [23].
Bipolar coordinates were also used by Podstrigach [24] to study the stress concen tration in an infinite el astic plate weakened by two unequal circular holes . He has studi ed two cases : (i) with the boundary of each hole subjected to uniform pressures of different magnitude, (ii) with the pla te uniformly stretc hed and the boundary of the holes being fr ee .
With the help of conformal mapping , stress distribution in a plate with two circular holes of different diameters was fir st analyz ed by Davies and Hoddinott[25] in 1963 , but this me thod is rather restricted . Their paper examines the accuracy of a simple theory which superimposes the solution for a small hole in the field of stress generated by a larger hole .
Other res earch work in this field has been by Hodden [26] who obt ai ned the solution in a closed form for unequal holes under uniax ial external force , with the same approach (1967) .
In 1968, Sale rno and Mahoney [27]  The integral equations involved in the above calculations have been throughl y studied by Sherman [28] who pro ved , a mong other things, that they can be solved by the method of successive approximations . So koln ikoff [29] in his book has considered the solution to the problem of eccen tric ring und e r uniform pressure and also concentric ring under conc entrated forces, wi th the help of the same method .
(c) Schwarz ' s Alternating Method : Since elastostatic p roble ms in multiply connected domains present serious co mpu tational difficulties , i t is natural to attempt to redu ce their so lu tion to a sequence of problems in simply-connected domains . This can be done by making some rather obvious modi f ications in Schwarz ' s treatment[30] of the Dirichlet problem for t he overlapping domains .
The proof of convergence of the Schwarz algorit hm in the * While obtaining the transformed stress functions with the hel p of equations (88~6 ) in [29] , Salerno has interpreted the m in a wrong way . The value of r i n the third term is the boundary value , the radius of t he hole, which he took as a vari able rad ial c oordinate instead . Then because of the third term r 2 / z ( = z) the function is no longer analytic and hence cannot be differentiated and which he did by taking r 2 outside the diffe rentiation as a constant and thus the second mistake. 5 solution of the second elastostatic boundary-value problems for a doubly connected domain for the case when the contours bounding this domain are sufficiently far apart has been supplied by Mikhlin[31]. In essence Mikhlin ' s proof is based on Neumann ' s modification [32]. A detailed and careful presentation of Schwarz-Neumann method or solution of the Dirichlet problem for a class of elliptic partial differential equations in two dimensions and in solving certain systems of integral equations will be found in the book by Kantorovich and Krylov[33].
A general proof of Schwarz algorithm for the second bounda ry value problem of elasticity in three dimensions was sketched out by Soboleff [34]. This reduces the considerati on of convergence of sequences of approximate solutions, to a study of convergence of sequences that minimize the integral for the strain energy. and ' b+e:.a' being inserted in the two holes of radii ' a ' and ' b ' 6 respectively, where ' e ' is very small as compare to the radii and radius ' b ' is greater than radius 'a'. ' c ' is the center to center distance between the two holes. These two inclusions inser ted without any rotation produce different uniform radial pres sures ' p ' and ' q ' respectively at the hole boundaries ' a ' and ' b ', as shown in Fig . 1 .
The purpose of this study is to determine the stress distribution around the two unequal holes . iii) Plate material is assumed to be linear-elastic , homogeneous and isotropic .
iv) The displacements in the plate are infinitesimal so that the classical theory of elasticity is applicable .
v) The two ri gid inclusions are inserted without any rota tion and with no slip so that the shear stress at the hole bound aries is zero and they are assumed to produce uniform pres sure inside the hole boundary . This value is substituted in the equations (88 . 6) given in [29] to give the new stress functions which when added to the pre vi ou s t r an sformed ones give the second approximation . Same procedure is followed starting from the bigger hole of radius b and fi nding the second approximation with respect to the smaller hole ' a '.
On the similar basis successive approximations can be obtained . Radius of the bigger hole .
Center to center distance between the two holes.
Shortest distance between the two holes [= c -(a+b)}.
Difference between the radius of inclusion and corresponding hole .
Uni f orm pressure inside the hole of radius a .
Uniform pressure inside the hole of radius b .

Inclusion.
Origin of the polar co-ordinates and center of corresponding hole .  Radial and tangential normal stresses respectivel y in polar co -ordinates .
Shearing stres s in polar co-ordinates.
Boundary value of stresses .
Boundary value of complex number . Solu t ion w. r . t . 01: AP, BP , CP , DP, EP and FP are as defined~ on page 50.
So lution w.r . t . 02 : where Q, . is a con jugate harmonic function of P, , then r/J(L) and Xt~) are anal y tic functions.
Stress field in polar co -ordinates , in terms of these stres s functions, a r e given by a nd 12 where ~=/re~·&_ Equating real and imaginary parts of equation (2), two separate equations can be obtai ned and thus knowing the stress functions the three unknown stresses can be determined from equations (1) and (2).
Let S be a closed smooth contour . -"' and -11be the component of stresses acting on the small element "1 of t he contour , parallel to the x and y axe s respectively . The stress boundary condi tio n on S can be expressed in terms of the above stress functions as II-2 SCHWARZ ' S ALTERNATING METHOD: Consider a region( Fig.-2 ) f ormed by the overlapping domains R 1 and R 2 , each of which is b ounded by a simple clos ed contour. Let the portion of t he ' contour c 1 bounding R 1 that lies within the region R 2 be c 1 " ' " and the part that is outside R 2 be c 1 . Then c 1 = c 1 + c 1 . Si milarly, denoting the part of t he boundary c 2 of R 2 that is I n interi or t o R 1 by c 2 and the remaining part by c 2 , it can be ' " wr itten as c 2 = c 2 + c 2 . The region R 12 that is common t o R 1 ' ' and R 2 is thus bounded by c 1 and c 2 , while the region R 1 +R 2 13 " " has the curve c 1 +c 2 for its boundary , as s hown in Figure 2 .
Let the values of some function ¢ be spe cif ied on the " " boundary c 1 +c 2 and ¢ in the r e gion a 1 +a 2 is to be determined The operator L is given by the equation (3) a nd the boundary condition on c 1 +c 2 by To obtain the first approximation ( <A / x," ) to ( ¢ / X.

)
the functions ¢,; X,' in R 1 are determined , so that To get the second approximation ( ¢ 2 J x; ) , the solution in R 2 is considered , such that For the third approximation , the solution is dete rmined in R 1 , satisfying the condition = and so on .
The use of this procedur e in constructing the approximate solution of special elastostatic problems in doubly connected domains is presented i n detail in [29] .

II-3 . THE COMPLEX FUNCTIONS IN THE FORM OF COMPLEX INTEGRALS :
The simply connected region can be mapped conformally into 16 a circular region, with the help of a suitable mapping function.

Let it be
The boundary condition at ' t ' will now be Similar equations can be wr itten for the second origin o 2 by changing the suffixes to 2 from 1 in the equations (6 ) and i92 (7) and to b from a . Now on c 2 we have t 2 = b e These equations ,( 6) and (7), are similar to the equation s (88.6) on page 324 of [29] and it may be noted equation (7) does not correspond to its counter part, because of a typographical error in [29] which is corrected herein.

II -4 . CAUCHY INTEGRAL THEORE MS :
Integrals of the type which occur in equations (6) and (7) are called Cauchy Integrals . Simple formulas for several cases of these integrals are given below [3] .
Let L be a simple smooth contour as shown in Figure 3 .
Denote by S+ the finite part of the plate bounded by L, and by s -the infinite part of the plate bounded by points lying outside L. The positive direction of L will be chosen such that the regions+ lies to its left .
i) Let f(z) be a function, single valued, holomorphic in S+ and continuous at S+ + L (region S+ and points on L), then Substituting these into eq uations ( 6 ) and (7 ), the functions ~1 c ;!",) and x 0 : c~,) are obtained . With these values the f inal state of stress is calculated from equations (1) and (2) .
(a) Determination of ¢,c~,) and X/c7E,) : For the unpenetrated plate the external force obviously van ishes and hence the stresses are zero , leadi ng to the res ults , The pressure p can be decomposed into two rectangular where t, =a e ~·19; ' the points on the boundary of the hole, and also (14) Substituting (13) and (14) into e quation (6) and (7)   The stress function (17)  Substituting these into equations (6) and (7) to transform (17) to o 2 as shown i n Figure (5).
Since the stress components are not altered by trans lation, one can wri te ii) The boundary value at radius b : Substituting the transformed stress functions given by ( 20) and (21) Substituting the corrected boundary values from (22) and (23) into the equations (6) and (7) (20 ) , (21) , ( 24) and (25) into the above expressions , the functions ~c Zz.) and Xa"' ( JZ:a.) are obtained as and ( 27) These functions are summarized in Group I of the Appendix A.
(c) Boundary Condition Check at Radius b : In order to check the boundary condition at the hole with radius b , stress functions (26) and (27)

+ c
Simplifying this one can obtain (28) Thus the stress functions given by (26) and (27) satisfy the boundary condition at the bigger hole of radius b .

III-2. Commencing with the Stress Field Around a Hole of Radius b :
Now in the similar way t he problem is solved star t ing from the hole of rad ius b.
Considering the infinite plate , containing only one hole of radius b with uniform normal pressure q applied along its :/, Substituting these corrected boundary value s (31) and (32) i nto equa tions (6) and (7)
Thus the boundary condition at the smaller hole of radi us a is satisfied as constant pressure p by the stress function s ( 3 5 ) and ( 3 6 ) .

I II-3 . FIRST AND SECOND APPROXIMATIONS :
The symmetric stress functions g iven by the equations (17) and ( 29 )  The stress functions given by (26) , (27) and (35) Putting these correct ed bound ary value s from equations (39 ) and ( 40) into (6) and (7)  Substituting from (43) and (44)  Putting different values in equation ( 51) the tables 2 and 3 are forme d . Table 2 First Approximation ,  (1) and (2) to g ive the following re sults 43 for e 1 =o and e 2 =n , which are arranged in nondimensional form. Similar procedure is adopted for the radial stresses fro m ( 56) and ( 59 ) •

44
Putting different values for the variables in the above equations, following tables are calculated • .
It can be seen that the above equation s satis fy the particular case for on ly on e hole in the infinite plate . ( 55) and ( 56) and a = O in the equations ( 58) and (59) , it is observed that these equations reduce to (49) and (50) which are the results for the single hole solution . Table 4 Se c ond Approximation , for c =4 . 0 , p = q , b I a = 2 . 0 .  Table 5 Second Approximation , for b/a = 2 . 0 a nd c/a vary ing .

Substituting b = O in equations
where A and B are gi ven by (61) and (6 2 ) Table 1 shows the results from the equation (48 ) which are symmetric in any direction with respec t to the center of the hole, because onl y one hole is considered .
It is clear from equation (48) that the radial and tangential stresses dec rease as r /a increases .  ( Figure 9 ).
ii ) As the distance between the two holes increases the effects of pressures p and q on other edge , i . e ., G and F respectively, decreases .
iii) When b>a the effect of q on the edge F is more than that of p on G a nd as the ratio b/a increases the effect of q on F increases . In other words , if b>a the maximum stress concentration for p=q will occure at the inner edge of the smalle r hole , i . e ., at F .  (55 ) , (56 ), (58) and (59 ) and the tables 4, 5 and 6 follow ing observati ons are made : i) As the two holes come closer the maximum stress concentration of the tangen tial stresses at the inner edge of the smaller hole increases .
ii ) As the two holes come closer stresses along the line FG increase rapidly whereas the eff e ct is not significant beyond the points E (along e 1 =n ) and H ( a long e 2 =o ) .
iii) As b is greate r than a the effect of q on the ed ge F is more than that of p on G. Hence the tangential stress concentration is at the point F . iv) As the ratio b/a increases the maximum stress concentration for the tangential stresses also increases .
v) For the case of b =a the results wi t h re spe ct to the two ori g ins o 1 and o 2 are found t o be symmetric .
vi) Fi gures 17 , 18 , 19 , 20 , 21 and 22 show the stress distribution of the tangen tial st resses around the two holes .

72
They also show the change i n st r ess distribution with respect to the change in the pressure ratio p/q or the radii of inclusions .
vii) The shear stress along the line of symmetry , passing through the points E , F , G and H~ is zero .
~iii ) Figures 27 and 28 show the stress dist r ibution of the radial and shear stresses around the edge o f the two holes .
These stresses should be of the magnitude p or q and zero respectively , but they do not satisfy these condition s exactly .
However , these figures clearly show that the second approximation is fairl y c~ose t o the exact solution .
The third approximation may give closer solution to the exact form [27].
2 . With the similar approach of successive approximation to the complex stress f unct i ons by the Schwarz ' s Alternating method, following problems may be tackled: i ) Infinite plate with three or multi holes .
ii) Inclusions with uniaxial or biaxial stresses at the infinity .
iii) Inclusions o f elliptic shape, square shape or any arb itrary shape.

= .
Ir --+ ~ --+ · · · 1 dt".z Second , third a nd all other higher or der terms in t 2 are 81 analytic in S+ and hence from equation (8)   . . c"-_ a.,._ (C +4} (C -a.) ~~etc . are all analytic ins+. Hence from the equation (8)  [ k, +1<.zJ where K 1 and K 2 are the residues at the Second and Third poles as stated above. Here f (t 2 ) has four poles, of which only one is inside S+ , at t 2 = 0 and other three are in s -region. Hence integrating inside s+ , one can write from equation (10) 26 } Here f (t 2 ) has four poles, of which only one lies i nside S+, at t 2 = O and it is of second order . The other t hree are . in sre g ion . Hence integrating inside S+ re g ion one can write from equation (10)