Karunes, B

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ON TH E DISTRIBUTION OF INITIAL STRESS DUE TO DISLOCATION IN AN INFINITE fL A T E  CONTAINING TWO UNEQUAL C IR 0)L A R  HOLES*By B. K A R U N isDepartment op Applied Physics, Univursitv Col|k(;e ok Sciknck and Technology,87Calcutta f{Received for publication, Afiril 1 7 , 1 0 2^)ABSTRACT. Solutions are obtained for two problems on state of initial stress due to dislocation in an infinite plate containing two unstressed unequal circular holes. The dislocations are due to (i) a parallel fissure joining thi two circular holes or joining each hole to infinity and (2) two opposite kinds of wedge shaped fissures of the same apex-angles, on two sides of the it'axis. Cases of equal holes in each of these problems are treated as particular examples and tables for numerical values of the constant coefficients when the hole boundaries are a=o.<S and o = -o .8 are given. Oraplis showing the stresses are drawn and diwscussed.I N T R 0 I) TT C T 0 NLet an elastic body, occupying a multiply connected region, be cut along a system of barriers which would make the region occui)icd by the body simply connected and let the two faces of each barrier be rejoined after removal or insertion of thin slices of the same material. Then the state of initial stress of the body is given by a solution of equations of elastic equilibrium which gives rise to multiple valued displacements, the displacements on one side of the barrier relative to the other being possible in a rigid body. In the present paper solutions are given of two problems of dislocation in an infinite elastic plate containing two unequal circular holes. In the first, dislocation is due to a parallel fissure joining the two circular holes or joining each hole to infinity, while in the second, dislocation is due to a wedge shaped fissure on one side of the :v*axis with its apex at the origin and insertion of a similar wedge shaped strip below the .v-axis. The solutions are obtained in bipolar co-ordinates, defined by the substitution (Jeffery, 1921)x + iiy-a)Here fl sin/? asinhexco sh a-co s/3 ’ cosh « - cos ^and j _ d(<x + iP) _ cosh « -  cos H d{x + iy)* Cournmniceted by Prof. P. C. MahantiIn terms of the stress function x> the displacements ate given byi / f l  B. KarunesJ L . k ,A + /X 0a 9/3fi , 0A , QQ A + // 0/3 0awhere2^ A + /a)and the stresses are given by aaa = (cosh « -  cos -  sinh -  sin j8 ^  -r cosh a | (h \ ) ^ 0^  0 0 ' j  (cosh < n-cos^ )~  -  sinh ~ sin + cos ^ ^  (fexiaa^ = -  (cosh a -  cos... ft)(21•■ • 3^)D I S L O C A T I O N  D U E  T O A I> A R A L L E 1/ F I S S U R ElyCt a = « i and « = - a .  be the boundaries of the two’ unequal circular holes in an infinite'plate.To obtain many valued terms in expressions for displacements, let us choose a stress functionhXn — A» sinh a (4 )Writing only the many valued terms in the displacements, we haveK sinh «sin ^3/u{A + fx) ’ cosh <x “  cos ^A + 2/i2jJL(\ +  fx)A/3(i — cosh a cos /3 ) cosh ot -  cos is(5)when /3“ jr _ A + 2/1 . 2/a(A + //)‘ (61and when /3= _ A - 2/» 42/r(\T/i)' ^Thus u remains continuous across the barrier /3= ±ir, i.e., the y-axis and V suddenly decreases by the constant amountA. + 21*V .  vA^(A+m)Therefore, the chosen stress function hxo will suit the state of dislocation due to a parallel fissure of that thickness along the line joining the centresDisMbution of Initial Stress, etc. 3 t9of the two holes or any other line parallel to this line cutting both the circles ot=«i and « =  - « j .The stress function produces normal stresses over the circular boundaries and «=  — and they are gi'i^n bya  a « i=  ~ A  sinh ; a a«_2* - A  sin/i*o(j (8)To reduce these stresses to zero, for the (|pise of stress-free boundaries, add a stress function h x t  to h ^ „ ,  where (/»Xi='BoiW(cosh <x-cos cosh 2a + jpji + C,i sinh 2«)cos (9)and it gives stresses over the circular boundaries, equal and opposite to those given by fcx«, i - c .  |Aacx, = —  sinh ; ««_a sinh (10)From the solution given by JeSery (1921) for an eccentrically bored pipe under constant pressure over its boundaries, where he used the same stress function as h ^ i ,  we obtain by suitable substitutionsBfl, =  2/4M(sinh“<X3- sinh “ot,) cosh {a, +«„)A i i =  -  JM(sinh sinh “«,) sinh («i-«2l C'i, =  ,4 M(sinh ^o^„—  sinh *«,) cosh ( n ,  - a . j )  ^Bn = /4M{sinh ’“a, cosh (a, + otj) sinh 20(2 + sinh ‘*«2 cosh (a, + otj) sinh 2«,-fsin h  “«2+ sinh “«,) sinh (a, +«2){In )whereM = i cosechfa, + Wa){sinh“«2 + sinh (12)From the sum of the stress functions h ^ o  and h ^ , ,  we get no stress over the boundaries « = « ! and « =  - « g ,  but a constant all round stress is obtained at infinity («=o, P — o ) .  The value of this stress is given byo « « = a j8;8 ’ • A M  sinh «i sinh Kgfsiuh 2 («i + Wg) ~ sinh a«, — sinh aa,}=constant=S (say) ... (13)For the complete solution of the problem we must add to (/iXo + ftYi) another stress function fox* such that h x t  produces stresses equal and opposite to ' S j a )  at infinity and no stress over the boundaries « = « , and « = —«2. To get the required all round constant stress at a great distance from the holes, we may chooseAXao = ”  iS(cosh « + cos f i )  ... (14)and add to it another stress function+ log (cosh a —cos ^)^(cosh « —cos^) +2^0«(oc) co sn B j(is)wheremfor n ^  2, andcosh (w + i)« + B, cosh (n -i)a  + Cn sinh (« + i)« + Dfl sinh0,(<x) = /l, cosh 2« + B, +C'i sinh 2«B. Karunes I■ . (i6)hXai is such that i^  produces no stress at infinity and the sum UiXt<i +  h x u )  — hXz produces no stress over the holes (X -cXi and cx= -cXj.We get, when o( >  o,li')(2 -  "  isj^cosh (X h (B„ 4 /\ )cx cosh a -  K  log 2 cosh 'x + K e  + 1 i-(B „ -}*/\ )cx + /v log 2-2/v cosh <Ke *"“ + — c~ “^ + 0j(o(j| cos /3t»*2n(u - t;1 I00 1X  t^i(«)cos n/i w«2 Jand when <x <  o,h\2 ~ ~ iS ‘j^ cosh « 4' (B„-/\ )(x cosh a -/ v  log 2 cosh ck-^Ke°’(1 7 )(18)+ | i - ( B o-/\)(X4' /v log 2 - 2 /\ cosh cxe^ -^ + 0i(a)|cos jS“*■ 2  - ; ■■■ J--- r{ in + i)c (i? “ |cos n/^♦ «»2n{n ~i) I )+ '^2^<i&«(cx) cos n^]To calculate the values of the constant coefficients, the boundary conditions for no stress (Jeffery, 1921),and 1^9)^ ( h x )  ^  constant -  />6(Xa+(r(cosh a cos j9 ~i)4-t sin $ Jare applied separately on equations (17) and f 18) for the boundaries «=:«, and a== -“<Xa respectively. We obtainBo=JFCNjjcosh 2(ai + cx3) -i}(cosh 2(Xa““ Cosh 2<Xi)Ai -  -XNi(cosh 2<x,-i)(cosh 2«2~i)(sinh 2«2+sinh 2<Xi)+X/2 Bi = A'Ni(cosh 2a,-i)(cosh 2<X2~i) sinh 2(oCj + cXj)-2 ,C^  -  -ifC]V,f(cosh 20C3~i)(cosh 2<Xi-*cosh 20(2-sinh 2<Xi sinh 2<X2 ~ sinh^aaj + (cosh 20c, - 1) (cosh -  cosh 2^ 2 + s ■ ' 1 2CX, sinh 2aa+sinh^ aoti) ]Distribution of Initial Stress, etc. 321whereNi = {cOSh 3(«i + 0C2) —l}~*(cOsh 2«1 + C0sh 2«J —2)~* ••• (21)and for n ^ 2 ,A n — j^(nsinh 2«2-sinli 2«<Xj){ —(h+ i ) + m cosh 20(i+ cosli 2n«i}+ (« sinh 2« i-sin h  2n«i){-(« + 1 ) + « ^osh2«s + cosh 2w«j} + N"^j iv iv rj^ (n sinh 2a2 —sinh 2 n c x 2 ) { ( n c o s h  2<x,-Hcosh 2waj}+ (n sinh 2«i —sinh 2wai)}(n~ i) - n  ccSh 2«a + cosh 2mcx2} + j  (22)C‘« = KAT. ^  . r«{cosh 2(Mai—cx2)~cosh 2(ncX2-cXj)}n\n+1} L+ (n +ij{n(cosh 2« i—cosh 2CX2) — (cosh 2nai-cosh 2nof.n)] jDn— |n{cosh 2(ncXi+ CX2) "COsh 2(naa + <^ ij}w (n -i)  L-{w -i){«(cosh  2a i-co sh  2CX2) + fcosh 2 n(K^ -cosh 2 noij IJwhereN =  {cosh 2n^ <Xj 4-0(3/ -n *  cosh 2(0(1 4- a^ ) 4- —To obtain the values of K , we have00"S #»(o)~on = las fe\i is to produce no stress at infinity (a = o, P — o).PuttingAi = K/2 + Kai —2 + Kbiand for n ^  2KAn -  -  +Ka,n(n + 1 )B „ = _  +Kb,.n{n — i)we get from (24)K(a, + bi) + K  ^  (a« + fc„) = 2t i-2(23)(24)(25)a „  6|, an, being known from (22), the values of K  can be calculated.Therefore, the complete solution for the state of initial stress caused by dislocation due to a parallel fissure between the unequal circular holes is given by the stress functionB. Karunes^X=hXo + h\i + hXi= i4asinh « + B„i«(cosh a - c o s  + (/4,1 cosh 2« + I + 6' 1, sinh 2«) cos ^ (26)-  isjcosh  a +■ cos |8 + {/}„a + K  log (cosh « -  cos ^)}(cosh « -  cos ^+ ^ 0 « {a)co sn ^ ln • 1 JThe circiiiiifeiential stress over the hole boundaries <x —otj and o(= — ot, are calculated from the complete stress function hxa/8j8j =^ 2/1 cosh cXj(cosh (x, -  cos /3)+ JAM(cosb -co s  /i)tsinh‘^ cx3-sinh“ cxj X {sinh (cXj + cxj cos fi + sinh cosh (<Xi + «a)}-^ 5 (c o s h  (X,-cos/3)j^ +2iB„ + K) sinha,+ 0i "(«i) cos j8 -  2 K sin^/i(cosh a, — cos j8)"^i? 1+  ^  +  n“0 n ( « l) - ? n ( « i) }  cos In-2 Jand■ —2A cosh «2(cosh «2 -  cos—JAMicosh «2—cos iSjCsinh^ cXj —sinh“<Xj;X {sinh (a, + 0(2) cos /J 4 sinh «2 cosh (a, +«2)j-iS (co sh  «i-cosi8;|^ 2K c” “2 -2 (B „-/f)sin h  «2+ ^i"(-«2)cos/3-2/v sin®^ ’(cosh s ~ c o s  j8)“ *00 -1+ '2  {?*»"(-«2) + n > « (-o (,)-^ « (-a j)}  cos «/3 **•2 JIf the two holes be equal, i.e. « = «, and « = —«,, we havefex= ^ *sin h «  + Bu cosiS“  JSj^cosh « + cos B + /C(cosh <x—cos B)log{cosh « —cos B)+ f ,(« )c o s« /il «•! Jwherefor n ^ 2 , and andn^{(*) = An cosh (n + i)a+Bti cosh (w - i)a  ^ i M - A ,  cosh 2« + B,S — A sinh“ «,(27)(28)(29)(30)The values of the constant coefficients areBii = ^  sinhVj= tanh a,)Bi = IK  tanh cosh 2«, — 2and for n ^  2,An =  — [ i -  2<Xi+oosh 2n«i~(n + i) 1w(n + i) L sinh 2n«| H-H siiih 2«i JB„ = Ft + ” cosh 201, -  cjish 2W0(, -  (w - 1 )  1n(n - 1 )  L sinh 2««,,+ « sinli 2«, Jvhile K  is determined from the equationDistribution of Initial Stress, etc. 321(32)K = 2 (cosh 2«, - 1 )  ^ (cosh 2I1«| -  jt) - »l^ (cOsh 2«1 -  I) , .2 sinh 2«, w(n“—I Hsinh 2nai + « sinh 2a,)The stresses over the circular boundaiies « = «, and a -  -  a, are given by =a/3^_i= 2/1 cosh «i(cosh cos /?) —i 5 (cosh « i -  cos B)X [a/v cosh «j + 4 / I,  cosh 2«i cos /9 -  2/C sin“ /8(cosh a, -  cos /8)~' ... (34)40+ 2 2n { ( n  'f cosli in  + i)a, + (» -  i)/i« cosh In -  i)<x^ }cos n f t ]nmQThe numerical values of the constants and for several values of(X, are given in the Tables I, II and III.T abi.k 1»1 0.6 0. S 1 .0 1.2 1.4 1-6i1.8______2.0i2.2K 8  3 ,s8  I 2 -6 5 5 1 .4 8 5 0 S89 <■ ^555 <^ •^355 0.231 !i I.S2 0 101'rABMi II«1 0 .6 o.S I . O 1.2 1 1 1 61I s 2.0 2 2T .2 4 0 0 .4 4 6 0 177__0.074 0.032 0.014 1 f) ooA 0.002 0.001A t 0.243 0 .0 6 6 O.OTO 0.006 0,002 0.001A i o.o6d O .O IJ 0,0020 .0 1 5 0.0020 .0 0 4A t O.OOTm324 B. Karunea T able IIIStresses over the circular boundaries a=o.8 and « = —0.8 are shown in figure I, in multiples of /I/a, between /9=o° a n d 180" only, becauseF ig . Ithe stress distribution is symmetrical about the y-axis. The maximum stresses are, as expected, at /3= ± 180” i.c. at those points on the holes which are nearest to each other. The maximum stresses have the value 3 6 3 /I/a. At /3= o the stresses are negative on both the circles. The stress change from negative to positive at ^=^±37°, i.c. at about the middle of the top quadrants of the holes.D I S L O C A T I O N  DUE  TO W E D G E  S H A P E D  F I S S U R ETo obtain such many valued terms in expressions for displacements so as to suit the case of wedge shaped fissure between two uneqal circular holes in an infinite plate, choose= cosh «Distribution of Initial Stress, etc. The many valued terms in the displacements are given byu —  ^ ^ cosh « sin ^2/*(A + /k) cosh cx — c o s /3 I---- A + 2/i ^  ^ sinh <x cos /?2 f i { \  +  n )  cosh«— cosiSIf a >  o, we have when 13=n325(36)_ A + 2/1 . linh aM+*“ U , ----- T—- —“ 7T/I ----—2/x(A+/z) cdsha + Tand when 8^ = - ttA -f* 2Mi  ^— o\ — siiili a2/^(A +  /u) cos1i (X-m  The discontinuity over the barrier ±n is+ 2/4 . sinh a / v ^ -Vo (say)(37)(38)we have, when /3= ±»rThereforeAnd if (X <C o, we get/x(A + /x) “  coshoc-f I_ _ a sinh <xy= yo= - . ,cosh a + 1A + 2/4Vo = 7T/I//(A + /a) ‘ a yo__ A -f2/t4 7Ti4fi(X + fji) * 0. - yo(39)(40)So the chosen stress function h \ o  will suit the state of dislocation due to the removal of a wedge shaped strip bounded by the planesA+2ia 7tAl —TT-: y . —  • y /x(A + /x) a(41)above the x-axis, and insertion of another wedge shaped strip of the same description below the A-axis, or v i c e - v e r s a .The normal stresses on the holes produced by h^o  are given bya <xcXi sinh a 1 cosh (Xj), a«a_2= sinh (Xj cosh cx^ ) ... (42)To reduce these stresses to zero, a stress function feXj of the same form as in (9) is added to h ^ o .  In this case has to produce stresses over the circular boundaries as given byaacXj = si nh cXj cosh <Xj), a<xcx.2 ~^(^ 2““ a—1802P—7Adjusting the constant coefficients to suit this requirement, we getB«, — + -  sinh 2<x,— sinh 2«j} cosh («i + «j)Aj t = - i A M { 2 {c^ i + « j) -  sinh 2« i -  sinh ‘ a j  sinh (oii—«j)'C i j - i A M {2 (^ 1  + 0 2^) -  sinh 2«i~  sinh 20I2} cosh (a i-a*)Bji = i/lM[2(«2 sinh 2<Xj —«j sinh 2*2) coshfaj + Wj)+ {2{«i -<X2) -  sinh 2«i+  sinh aoCj} sinh («i + aj)]326 B. Kamme^• (44)whereM = i  cosech («i + «j)(sinh* ex, + sinh“ «,)-1 — (45)As in the previous case here too we get a constant uniform all round stress at infinity due to the stress function The value of this stress isi4M[(2«, -  sinh 2«,) cosh sinh « j-  (2 a - sinh 2«2)sinh a, cosh «2 + («2 sinh 2«, -  «i sinh aotj) cosh «i + «*) ]e=a «« —a fi^  =S'isay)(46)To leduce this stress at infinity to zero, a stress function h^ 2 of the same form and nature as in (15), with the substitution of S' in place of S, is added to (h \  + h^j). The complete solution is therefore given bykX = hX, + hX, + hX2= Aa  cosh a  + B„i «(cosh « -  cos /3)+{^,1 cosh 2 « + B „ +  Ci, sinh 2a} cos jS .. ( 7^)- iS '[c o s h « +  cos^ + {Bo« + /Clog(cosh« -  cos^)J(cosha-- cos/3)+ S  0«{<x)cos njS]«-iThe circumferential stress over the circular boundaries are now easily calculated from (47) and (3) as follows.over « = « ,a BBi =2(cosh «, — cos +B„,) sinh cosh 2«i+ C,i sinh 20tj) cos/8} , - iS '(c o s h « ,-c o S jS )[2K'e~'“ + a{Bo + K’)sin h « ,+ 0 ,"(«,)cosB -- 2 K  sin* B(cosh «, -  cos B)“ *+ 2 + n * 0 « ( o t j ) c o s « / 8 ]DisMhution of Initial Stress, etc. 327over «■ » -« 3a ^ ^ - a  =  -  2(cosh «a -  cos f i ) { ( A  + B„,)sinh + a (/ lc o s h  2«2- C i i  sinh awa) cos /9}— iS '(co sh « a - cos/J)[2Ke~**-2(Bo-h')sinh«j+0i"(-«a)cos/3 •" 4^9)-  aK sin* /3(cosh 0I3 -  cos jS)“ ^+ S  —« j ) —«a)jcOS tl ]^»i=2 yIIf the holes be equal, we have Bn = o, / l i ^ o  and S '= o . Thereforeh ^  =  A a .  cosh « + B, ,«(cosh a -  cos p) + C, i sinh a« cos ^ ... {50)where  ^ .B o i - 3 A M i 2V.i -  sinh 2<x,)cosh 2«j, C u ' — A M ( 2 0 > i  -  sinh 2«,) ... (51)The stresses over the hole boundaries are now given bya  = -a/3)8_i = 2(cosh«i- cos/3){(^+Bfc,)sinh«i-aCiiSinh 2«jCos^}... (53)/eo6 0  120Values of i8 in degreesF i g . 2In figure 2, the stresses in multiples A j a  over the hole boundaries <x = o.8 and “*0.8 are shown in a graph. Here again the maximum stresses are328 B. Karunesat i8= ±i8o® and the uiinima are at /3=o. The minimum and the maximum values are o.so6A la  and u .s g A l a  and are of opposite signs on the two holes.A C K N O W L E D G M E N TThe author wishes to express his respectful thanks to Dr. S. Ghose of the Department of Applied Mathematics for suggesting the problems and for his constant guidance during the preparation of this paper.R E F E R E N C EJeffery, G. B., igai. Phil. Trans. Roy. Soc., A221, 265.

On the Distribution of Initial Stress Due to Dislocation in an Infinite Plate Containing Two Unequal Circular Holes

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On the Distribution of Initial Stress Due to Dislocation in an Infinite Plate Containing Two Unequal Circular Holes

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