Bhargava, R D

Gour, P C

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CIRCULAR HOLE UNDER DISCONTINUOUS TANGENTIAL STRESSES*R. D. BHARGAVA ai0  P. C. GOUR**I ndian I nstitute of TECHNOik>GV> K anpur, I ndia{Received November 9, 1968)ABSTRACT. In this problem, the stresses «tkI flisplacements due to discontinuous tangential stresses applied in a specified manner at the boundary of a circular hole in an infinite plate have been studied. It may bo noted that the boundary tractions have a resultant. The problem has been solved with the help of integral equation method coupled with Fourier series representation of the boundary conditions.Consider a circular hole in an infinite plate and let tlie tangential stresses be applied at the boundary of the hole in the manner as shown in the adjoining figure 1. Let the radius of the circular hole be Jt with its centre at the origin so that the equation of the boimdarj  ^ of the hole may be written as <ro- =  1, whore <r is tlie complex coordinate of any Itoundary point.80As is well-known the solution of any two dimensional problem in elasticitycan be obtained if two analytic functions ^{z) and t(z) of the ^are known. The boundary conditions on the circle may be written as (Muskheh-shvili, 1963 and Sokokikoff, 1956)* This problem was referred to one of the authors (R.D.B.) by H. L. Oex, F.R.As London,♦ ♦  Engineering College, Dayalbagh, Agra.717or taking its conjugate, the boundary conditions are7 1 8  R. D. Ehargava and P . C. Oourwhere= / i — t /g  =  // l + * / g  = f  — i /  {Pnir+*^ny)^ . .  (1)and Pnx, Pfiy ®re the components of boundary tractions in positive x and y direc­tions on the surface whose outward drawn normal is n ; S is the arcual distance of a point on the boundary, measured from a fixed point.One has to be careful at this stage, d is the angle which the outward drawn normal (in the present case, this normal points towards the centre) makes with the rr-axis in the anti-clockwise direction. It is found that, if T be boundary tangential traction,=  T  Sin Oy P„y — — T  Cos 0, o < a < ^ ;Pfix =  — 5T Sin 0, P „y =  T  Cos d, IPnx ^  — T  Sin 9, P „y T  Cos 0,Pfix “  P Sin 9, Pny =-- - T ’ Cos a. ^  < a < 2 n .Substituting these values in (1) it may be seen that/ i +^/2 =  iRT[\—e^% in the first quadrant,=  e*^ ], in the second quadrant,=  2^+c'^], in the third quadrant,=  e*^ ], in the fourth quadrant.The function is expanded in terms of a Fourier series.I fwe easily obtain that/i+ i /a  — Sfor 1Gircnlar hole under Discontinuous StressesThe values o f some o f the non-zero coefficients are :7 1 9«o = iRT .TT -2-2i7T], «i -2iRT n ’ «a =  0,=: 2iRT 3tr ’ «4 =2iBT ~3F ’2iRT ftn ’„  URT “ * - -3 ..fs  ■a, = 2iRT in  ’2iRT In '2iRT "9s ’„ _  Slier 5-9s ’2iRT lln  ’ 1^2 =-2iRT - 2iRT T3s ’„  i2iier , “ “  ^  7T3s- ’Also,=  — 2iRT n ’ W-2 =  -URT— — , «_•. =  3s •*2iRT 3s ’„  _  2iRT “* 5s ’«-8 == —2iRT 5n ’ — —HiRT 2iRT in  ’„ 2iRT "  9s ’=  — 2iRT 9n ’l2iRT _  5‘l l s  ’2iRT 11s ’, 2iRT-  -  i3 „ .®-18 =  —2iRT IZn ’16»iJT7-15s’2iRT Ibn ’„  _  2iRT 17s ’etc.In the case of simply-connected infinite region <j>(z) and rjr(z) are given by Muskhe- lishvili, (1963) and Sokolinkoff, (1956).^{z) =  TRz- logf{z) ^  T’Rz+ log*+?^„{a). . .  (2)where Z is any point in the region; F and F' are the external stresses at infinity;is the resultant force on the interior boundary; and ^(z), t^f^ z) are two ^ ly t ic  functions- in the infinite region.In the present problem the stresses at infinite are zero, hencer =  r' =  0Also JC =  0 and y  =  4 RT. At this stage it may bo noted that X  =  §  Pn»d« aa Y =  ^  Pnf/As. As the outward normal points to the centre, a is to be measured in the clockwise direction.Substituting the values of F, F', X  and Y in (2) the above equations yield2iRT7 2 0  R. D. Bhargava and P . G. Oour=  -n ( l + K )log Z+^oi^),At this stage, it is convenient to write z =  Sp, thus mapping the circle of radius B in the 2^ -plano to a circle of unit radius in the p-plano. It is trivial to speak that the outside region in the z-plane is being mapped to the outside region in the p-plane.The functions <po{p) and ^o(P) corresponding to <f>Q{z) and ^q(z) may be obtained as follows (Mushelishvili, 1963).12mf /o i ?J <r—p ’V.. (3)12m \ < f - p  p..  (4)whereAlso /  is given by/ o = / +  ^  logo--n2iRT7r(l-fiC)Substituting the value of /g in (3), it is seen thatThus/k i rt\ 5  / I \«*.X ^^ o (P )-  —  1) 2»(2»+1)»r(l+X) n n-i 2»t(2»4-l)’ P** ' (5)Gircvhr M e under Discontinuous StressesTo calculate f (p ) ,  the value of ie calculated by7 2 1/o  — /■ 2iRT logo*+ 2iRT 1w(l+-K) 0-2whore_ ao ^/  =  S a„e<«9+S0 X ^S u b s t ifl^  the value o f i n  (4), it is found thato^(p) = f . i,+  s 2n+i 1 1^  L 2 (1 + ^ )  p2 > {2n— l)2 n '  p*» j 'Thus^ip) = \ - t t ^  -  + S  ( nn+i 2 «+ l 1 1^rd+Jf) XX l2 ( l+ i f ) > *  +  „ t j “ )^ r2n-T)2nV»J-Now taking.. (6)ci> (p) B<i>(p) 2iT r 1 1 «  1 1 1“ F 'L  r + x  p +  • 7;*»+r ]>or. 2*2’ r 1 J8 , £ , ,, 1I T+rz +■^- +r- ( f )  ]■ ■• mAgainOrn z )  =  * (p )=  ^f ( « )  =  2 » r r _  K  1 I 3+N  1 £ ,  , . .2 n + l  1 norNow taking * =  re*® and z =  re~*®, we obtainn - \ - d d  =  4He$(z)722 B. D. Bhargava and P. C. Qbur-  f  [ -  T T £  > f - . ' - * ' " " ’  H + T  (t /  “ “  ■ ■ ■ « ’>And0d—rr-\-2ir0 =  2[Z$'(z)+\Ji'(z)]e®*®_f / 1 -®  3 + A  ^  -® fitn L \ 1+A 7 r+A' r» / 1+A rSeparating real and imaginary parts from (10) it is found thatrr i 2T r 2K /R RH . .  , R /, (7t [ 1 + A  ( r r® ) **'*^  ^  r ( r®)sin d^ 1 + 2 ^ C os2 0 + ^+ i ( i  +  I )  } ]  'It , d . .  ( 1 1 )^  2T T 2K /B ,B h  . „ ,B  /, IP\i IT  I r F r ( 7 + 7 t ) “ ' ' + 7 ( l - 3 )Bind2m ml + ^ c o s 2 0 +  ^25T r 2K )B R*\ o , ^ l ^  ^*\ oo8<^[ - w ( 7 - f r ). (13)Circular hole under Discontinuous StressesOn the boundary o f the hole where R ~  r, tlie results are : rr — 0 ,ST r K66 =  T  [r+ J f +723rO =  —T. (14)Because o f discontinuities at 0 — ± 7 t/2, the normal stress 66 exhibits infinity at these points.For the displacement components we note that2/i{VrjrWg) =  e-^^[K<p{z)--zf(z)-Jrf(z)] (15)The displacement components are given by substituting the values of ^(z) and ijf{z) in the above expression.R E F E R E N C E SMuskhotishvili, N. I., 1963, Some Basic Problems of the. Mathematical Theory of ElaslkUy.P. Noordhoff Ltd., Gromingen.Sokolinkoff, I. S. 1956, Mathematical Theory of Elasticity, McGraw Hill, London.

Circular Hole Under Discontinuous Tangential Stresses

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Circular Hole Under Discontinuous Tangential Stresses

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