Two dimensional polar coordinate system in airy stress functions

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Frank Bartholomew Allen
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1 I J C T A, 9(9), 6, pp Intentionl Science Pess Two dimensionl pol coodinte system in iy stess functions S. Senthil nd P. Sek ABSTRACT Stisfy the given equtions, boundy conditions nd bihmonic eqution.in ode to solve -dimensionl iy stess function poblems by ug pol coodinte efeence fme, the equtions of equilibium, definition of Aiy s stess function nd one of the stess equtions of comptibility must be ecognized in tems of pol coodintes. Keywods: Fom Ctesin Coodintes in to Pol Coodintes, Aiy stess function fo pol coodintes, Eqution in pol coodintes, A stess field symmetic bout n xis, A cicul hole sheet unde emote she nd Exmple. INTRODUCTION The min pupose of this ddess is to bing to the ttention of the wokes in Aiy stess function nd elted bnches of pplied mthemtics simple genel method of solution of sevel impotnt clsses of -dimensionl boundy vlue poblem. A -dimensionl pol coodinte study in ings nd disks, cuved bs of now ectngul s section with cicul xis etc.ug the pol coodintes is dvntgeous to solve in iy stess function... Fom Ctesin Coodintes in to Pol Coodintes To tnsfom equtions fom Ctesin to pol coodintes, fist note the eltions x =, y = = x + y, tn y x x y, tn y x x y, y x x y x y x y Assistnt pofesso, Deptment of Mthemtics, Vel Tech Multi Tech D Rngjn D sgunthl Engineeing College, Avdi, Chenni, Tmil Ndu, Indi, Emil: senthil986s@gmil.com Associte Pofesso, Deptment of Mthemtics, C. Kndswmy Nidu fo Men College, Ann Ng, Chenni, Tmil Ndu, Indi, Emil: cicesek@yhoo.in

2 434 S. Senthil nd P. Sek x x x x y y y y x x x x Similly (..) y (..) x y Fom (..) nd (..) (..3) The Lplce eqution is x y (A) x y (B)

3 Two dimensionl pol coodinte system in iy stess functions Aiy stess function fo pol coodintes The pol coodinte in plne elsticity poblems such s the stesses in cicul ings nd disks, cuved bs of now ectngul s-section with cicul xis, etc. Fo two-dimensionl pol coodinte system, the solution to plne stess poblems involves the detemintion of in plne stesses.the stess tnsfomtion fom the Ctesin coodinte to the pol coodinte is x y x xy xy y x y xy (..).3. Eqution in pol coodintes The iy stess function is function of the pol coodinte (, ) the stess e expessed in tem of the iy stess function The bi hmonic eqution is,, (.3.) (C).4. A stess field symmetic bout n xis Let the iy stess function be (). The bi hmonic eqution is d d d d d d d d (.4.) Ech tem in this eqution hs the sme dimension in the independent vible such n ODE is known s n equi-dimensionl eqution solution to n equi-dimensionl eqution is of the fom Let equi-dimensionl eqution is Into the bi hmonic eqution, we obtin tht the uxiliy eqution is = m (.4.) ( 3) = (.4.3) m (m 3) =. m = o (m 3) =. The fouth ode lgebic eqution hs double oot of nd double oot of 3. The genel solution is

4 436 S. Senthil nd P. Sek Whee A, B, C nd D e constnt of integtion. The components of the stess field e () = A log + B 3 log + C 3 + D (.4.4) A 3log 3 B CR A 5 log 6 B C (.4.5) The stess field is line in A, B nd C. The contibution due to A nd C e fmili they e the sme s the lme poblem. Fo exmple, A hole of dius in n infinite sheet subject to emote bixil stess S, the stess field in the sheet is S, S (D) The stess concenttion fcto of this hole is we my compe this poblem with tht of spheicl cvity in n infinite elsticity solid unde emote tension 3 3 S, S (E).5. A cicul hole in n sheet unde emote she The sheet is stte of pue she The emote stesses in the pol coodintes e xy s, xx, yy. (.5.) We know tht S, S, S (.5.),, We guess tht the stess function must be in the fom The bi hmonic eqution is, f (.5.3) d d 4 d f df 4 f d d d d (F)

5 Two dimensionl pol coodinte system in iy stess functions 437 A solution to this equi-dimensionl ODE tkes the fom Inseting this fom into the ODE, We obtin tht m f (.5.4) m 4 m m 4 o m The lgebic eqution hs fou oots,,, 4. The stess function is The stess components inside the sheet e C A B D 4, (.5.5) 6C 4D A 4 6C A B 4 6C D A B 6 4 Whee A, B, C nd D e constnt. To find the constnt A, B, C, D.We invoke the boundy conditions: (.5.6) S. Remote fom the hole nmely,, S, S, giving A, B.. On the sufce of the hole, nmely,,,, giving The stess field inside the sheet is D S 4. S s 3 4 s 4 3 (.5.7)

6 438 S. Senthil nd P. Sek Exmple.. A thin plte is subjected to unifom tensile stess t its ends; Find the field of stess existing within the plte. Solution: The oigin of coodinte xes t the cente of the plte The stte of stess in the plte is The stess function is y,, x y xy stisfies the bihmonic eqution. The stess function my be tnsfomed by substituting y =. The stesses in the plte 4 Exmple.. A thin plte contining smll cicul hole of dius is subjected to simple tension. Find the field of stess nd compe with those of plte contining smll cicul hole. Solution: The boundy conditions ppopite to the cicumfeence of the hole e, (..) Fo lge distnce wy fom the oigin we set, nd equl to the vlues found fo solid plte fo. We ssume stess function (..) f f (..3)

7 Two dimensionl pol coodinte system in iy stess functions 439 In f nd f e yet to be detemined d d d f df d d d d (..4) The solution of eqution is d d 4 d f df 4 f d d d d (..5) f = A log + B log + C + D (..6) 4 G f E F H (..7) Whee A, B, C, D, E, F, G nd H e constnt of integtion. The stess function is then obtin integting eqution (..6) nd (..7) into (..3) by substituting into The stess e found to be 6G 4H A log B C E 4 c 6G 3 log 6 A B E F 4 6G H E F 6 4 The bsence of D indictes tht it hs no influence on the solution. (..8) Accoding to the boundy condition () A = F = in eqution (8) becuse s the stesses must ssume finite vlues. Then, ccoding to the condition (), the eqution (..8) yield C 6G 4H 6G H B, E, E 4 4 Also, fom eqution (..) nd (..8) We hve = 4F, = 4B Solving the peceding five expessions, we obtin (G) (H) 4 B, C, E, G, H (I) The detemintion of the stess distibution in lge plte contining smll cicul hole is completed by substituting these constnt into eqution (..8) 4 3 4

8 44 S. Senthil nd P. Sek (..9) REFERENCE [] S.H.O, C. Hillmn, F.F. Lnge nd Z. Suo. Sufce ccking in lyes unde Bixil, Residul compessive stess J.AM.cem.soc 78[9] (995). [] S.P. Timoshenko nd J.N. Goodie, Theoy of Elsticity 3 d ed.mcgwhill, New Yok (97). [3] S. Woinowshy-kiege nd S. Timoshenko Theoy of pltes nd shell McGw-Hill, New Yok (97). [4] Lekhnitskii S.G. Theoy of Elsticity of n Anisotopic Body Mi Publishes, Moscow (977). [5] Poulos nd Dvis Elstic Solutions fo Soil nd Rock Mechnics Wiley, New Yok (974). [6] Two-dimensionl poblems in elsticity -chpte 3 [7] Sokolnikoff, I.S. Mthemticl Theoy of Elsticity (nd Ed.) N.Y.: McGw-Hill (956). [8] Englnd A.H. Complex Vible Methods In Elsticity London, Get Bitin: Wiley & Sons Ltd (97). [9] Dshini Ro Kvti nd Nomu, Seiichi. (5) Aiy Stess Function fo Two-Dimensionl Cicul Inclusion Poblems UTA Mste s Thesis. (5). [] Solution of dimensionl poblem in pol coodintes solid mechnics by ES4 (Fll 7).

Chapter 6 Thermoelasticity

Chapter 6 Thermoelasticity Chpte 6 Themoelsticity Intoduction When theml enegy is dded to n elstic mteil it expnds. Fo the simple unidimensionl cse of b of length L, initilly t unifom tempetue T 0 which is then heted to nonunifom