Fatigue Failure of an Oval Cross Section Prismatic Bar at Pulsating Torsion ( )

Transcription

1 World Engineering & Applied Science Journl 6 (): 7-, 5 ISS 79- IDOSI Publiction, 5 DOI:.59/idoi.wej Ftigue Filure of n Ovl Cro Section Primtic Br t Pulting Torion L.Kh. Tlybly nd.m. giyev Intitute of Mthemtic nd Mechnic, Azerbijn tionl Acdemy of Science Abtrct: The number of pulting torion, t which the firt dmge nd ftigue filure of n ovl cro ection primtic br hppen, re determined. The br' mteril h no trengthening nd the pltic re entirely cover the ection' contour. At the initil eltico-pltic torion of the br from the nturl tte, V.V.Sokolovki' olution w ued. Intenity of reidul deformtion i ccepted for determining ftigue filure prmeter. Key word: Ovl ection br Pulting torion Dmge Ftigue filure ITRODUCTIO ( ) ( ) x = + b in + bin co, Here, under ftigue or cyclic durbility it i y = b co + bco co, commonly undertood dicontinuitie of the mteril of where i lope tngentil to the ection contour, > contruction under conidertion under cyclic chnge of b Herewith, the emi-xe of the ovl (ellipe) will be: pltic deformtion [-]. At ome experimentl + b, - b. A qurter of the conidered ovl cro ection invetigtion [] crried out by infrred pectrocopy i given in Fig.. nd coutic emiion it i noted the fct tht firt ftigue Let the conidered br in the nturl tte be dmge of tructurl element pper fter certin ubjected to the torque M. Under the ction of the torque number of cycle fter the trt of cyclic deformtion M in the br there pper the tree, yz trin, proce. The number of cycle of loding, preceding to yz nd diplcement u x, u y, u z. The coordinte ytem formtion of firt dmge nd to filure re x y z, whoe origin coincide with the center of the ovl, commenurble vlue. Here, by uing the trt of the xi x with the xi of the br, the xi z with gret cyclic dmge condition nd cyclic trength obtined mll emi- xi, the xi y with mll emi-xi, i ued. in [5], problem of ftigue filure of n ovl cro The problem i in determintion of tre, trin nd ection primtic br t pulting torion i olved. diplcement field in eltic nd pltic domin nd Herewith, the trengthening of the br mteril i lo in etblihment of boundrie between thee excluded (idelly eltico- pltic deformble mteril i domin. The given problem, the problem of initil eltico conidered) nd it i umed tht the pltic re entirely - pltic torion of br under uppoition tht the br cover the br contour the known olution of mteril h no trengthening nd the pltic re entirely V.V.Sokolovkii [] i ued. In the unloding proce cover the cro ection, w olved by V.V. Sokolovky ppernce of the re of econdry pltic deformtion by the pecific, o clled the invere method. i llowed []. It i umed tht the hypothei of plne cro The reidul tre nd trin re found by uing ection hold. Therewith the diplcement u x, u y, u z the V.V. Mokvitin theorem on econdry pltic my be repreented in the form. deformtion []. u = yz; u y = ; u z = f( xy, ), () Sttement nd Solution of V.V. Sokolovky Problem on Initil Torion of Br: Let conider n ovl cro where i the twit ngle of unit length br re, f(x, z) ection (lmot elliptic) primtic br whoe contour i i torion function chrcterizing the wrping. Reltive determined by the following eqution in the prmetric torion ngle i conidered to be poitive for form []. definitene. Correponding Author: L.Kh. Tlybly, Intitute of Mthemtic nd Mechnic, Azerbijn tionl Acdemy of Science 7

2 xx = yy = zz = xy = b 5 b Since the tree nd G M = 5 + xx = yy = zz = xy = trin, then the equilibrium, (9) equlitie nd condition of trin comptibility ccept the form: b f ( x, y) = xy, () yz + =, () x y G( + b) G( b) = y ; yz = x, () yz + =. () x y + b b = y ; yz = x The Cuchy kinemtic reltion hould be fulfilled:. () The diplcement u x, u y, u z re determined from u u x y u y u ; z = + yz = + () formul () tking into ccount () nd (). y x z y In conformity to the conidered problem, the Since the lterl urfce of the br i tre free, on intenity of the trin t i expreed by the formul: the cro ection contour the tngentil tre vector hould be directed long the tngent to the contour: ( ) / t = + yz. () yz dy = = tg By uing () in formul () the intenity of eltic dx deformtion will be: The principl momentum of tngentil tree cting on the br cro ection equl the torque M : t = / ( + b) y + ( b) x. () ( x yz y ) df = M, (5) F Let for = on the boundry of the cro ection Here F i the re of br cro ection. contour firt the pltic deformtion pper. It i cler tht the twit firt will ccept the vlue t the point x In the eltic domin, the Hook lw hold: =, y = ± ( - b). At thi point, the yield condition (7) h the form =. Allowing for the firt formul of () we = G = G, (6) get: yz yz World Eng. & Appl. Sci. J., 6 (): 7-, 5 where G i her modulu? = ( )( ). In the pltic domin, the von Mie yield criterion i G b + b (5) fulfilled: The torque M t which pltic deformtion pper, bed on reltion (9) nd (5) re expreed by the / + yz =, (7) formul: 5 b 5 b where i the yield point in her expreed by the yield M = 5 +. = /. point in tenion ( b)( + b) by the reltion? Furthermore, the continuity condition of tngentil (6) tre nd xil diplcement component when ping For M > M, in the cro ection of the br there through the boundry of eltic nd pltic domin i rie pltic deformtion re. In the ce when the fulfilled. pltic re entirely cover the cro ection, it hold the According to [], the olution of the eltic problem V.V. Sokolovkii olution []. According to h the form: V.V.Sokolovkii' invere method, the eltico pltic boundry i given in the form of n ellipe b 5 b = 5 + M () G x y + = or c d

3 World Eng. & Appl. Sci. J., 6 (): 7-, 5 Herewith, for the emi-xi c nd d the following The tree nd yz hve the following formul re obtined: expreion in the eltic re. c b, = + + ± d Gb Gb / = G y + +, Gb G b (7) () Where the twit nd the torque M re connected with the formul: ( 9 ) M = b + b b + + G b Gb Gb / / yz = G x + +. Gb Gb (5) In the pltic re co, in. = yz = (6) Herewith, long ll the cro ection it hold the () eqution [6]: In the eltic re (inide the ellipe) the diplcement yin + x co = b in u i repreented by the formul: / u z = xy +, G b G b In the pltic re by the formul: / y = + +, Gb G b / x yz = + +, Gb G b yz = x, = ( y bco ). (9) / + +. Gb Gb b / Reidul Stree nd Strin of the Ovl Br: Define x u co z = b + + in now reidul tree nd trin tht re preerved in the b G b ovl br fter removing the torque. nd we will conider () tht in the unloding proce in the br there rie the re of econdry pltic deformtion. To thi end, we In the eltic re, the deformtion nd yz re will ue the V.V.Mokvitin theorem on econdry pltic expreed by the following formul: deformtion []. According to thi theorem, the reidul tree, yz trin, yz nd diplcement u x, u y, In the pltic re we hve () = ; yz = yz yz ; = ; yz = yz yz ; u u u u u u () (7) u u u x = x x; y = y y; z = z z; =., yz,, yz, u x, u y, u z, Here re the tree, trin, diplcement, twit, repectively before the beginning of unloding, tht re determined by the formul (), (5), () (), (),(9), (), (). The quntitie x z, y z,, yz, ux, uy, uz, re the () tree, trin, diplcement twit tht hold in 9 ote tht the olution of (9) (6) w obtined by V.V. Sokolovki [] nd i true provided c + b, or. u of the twit z my be determined by the formul:

4 World Eng. & Appl. Sci. J., 6 (): 7-, 5 fictitiou, ovl br t elticopltic torion by the me We define the reidul twit (M ) by the formul torque M. Unlike the conidered br the yield point t the ( M ) = ( M,, G,, b) ( M,, G,, b). Herewith the her of mteril of the fictitiou br i. Between the quntity i found from the eqution (), the quntity quntitie M nd it hold the reltion () by chnging from (). The ellipe eprting the re of eltic by : unloding nd econdry pltic deformtion re 6 determined by the emi-xe c nd d coinciding with the M ( 9 ) = b + b b + emi-xe c nd d repectively. Gb / +. c = b + + ± Gb Gb () d Gb ( ) Gb ( ) Herewith the eltic pltic boundry in the cro (6) ection of the fictitiou, br will be n ellipe with the emi-xe c nd d: Herewith the following condition hould be fulfilled: d - b (7) c b. = + + ± Gb d Gb By decreing the torque from M to M, there will (9) hppen eltic unloding. By decreing the torque from M to zero, the re of econdry pltic deformtion In the eltic domin, the cro ection of the extend from the point [, ± ( - b)] to the centre of the fictitiou, br will be: cro ection of the conidered br. Herewith, the externl / contour of the re of econdry pltic deformtion = G y + +, will the prt of the contour of the ovl cro ection with G b Gb () the emi-xe + b, - b the internl contour will be the prt of the ellipe with the emi-xe c nd d, defined by / the reltion (6) (Fig. ). yz = G x + + +, According to the theorem on econdry pltic Gb Gb () deformtion [] nd formul (6), the econd pltic deformtion in the unloding proce will pper in the / ce if. y = + +, 5 G b Gb b 5 b M M = 5 +. () ( b)( + b) / x yz = Herewith, the condition of ppernce of econdry G b Gb () pltic deformtion t totl unloding will be: 5 b 5 b In the pltic domin of the cro ection of the M 5 +. () fictitiou br we hve[7]: ( b)( + b) = co, yz = in, () It i cler tht ubject to condition (), the condition (7) will be fulfilled. At limit vlue Mlim of the torque M, the pltic re t x z = ( y bco ), yz = x (5) initil torion fill ll the cro ection nd eltic kernel degenerte into ome ection whoe length ccording to In eltic nd pltic domin the expreion will be formul (7) will be b. For the quntity M lim ccording to written imilrly for u, u lnd u. formul (), we hve: x y

5 World Eng. & Appl. Sci. J., 6 (): 7-, 5 Fig. : Ditribution of pltic deformtion re in the qurter of n ovl cro ection br t torion nd totl unloding.,. b ( 9 ). The re between the ellipe with emi-xe (c, d) Mlim = b + b (9) nd (c, d ). In thi re, ccording to formul (6) nd (), () nd lo formul () nd (), () we hve: Herewith, the vlue of the torque M hould be le thn of M lim. Subject to (9) nd inequlity (), the inequlity M M lim fulfilled in the ce if the ovl cro ection prmeter nd b tifie the condition: = yz = co, in. ( ) yz = y bco, = x. / co = + G y + +, G b Gb () () Condition () hold ubject to the inequlity: / [( + b)/( - b)] 5,65. Thu, in the br mde of idelly in yz = G x + +, pltic mteril with cro ection in the form of n ellipe, G b Gb ppernc of econdry pltic deformtion i poible only the ce if the rtio of it gret emi-xi to the mll () one doen't exceed 5,65. Subject to thi condition, the econd pltic deformtion t unloding will necerily rie if the vlue of torque will tify condition (). Clculte now the reidul tree nd trin. / y y b co = + + +, Gb Gb Herewith, we hould ber in mind the exitence of there (5) different re in the cro ection (Fig. ). Are of econdry pltic deformtion. According to formul (6) nd () nd formul () nd (5), in thi re we hve: / yz = x+ x +. G b Gb (6) () The re inide the ellipe with emi-xe c nd d. In thi re, ccording to formul (), (5) nd (), () () nd lo formul (), () nd (), () we hve:

6 / = G y G y + + Gb G b / G y + +, Gb G b / yz = G y G y + + Gb Gb / G x + +, Gb Gb / = y y + + Gb Gb World Eng. & Appl. Sci. J., 6 (): 7-, 5 (7) () ome fictitiou br of ovl cro ection t it eltic torion by the torque tht w pplied to the conidered br before unloding. According to whp h been id nd on the be of formul (9) nd () we write expreion for nd :, x z, yz, b 5 b G M = 5 + ( + ) ( ) G b G b =, = x, y yz + b b =, = x. y yz The reidul twit will be: = - where nd re determined through the torque M by formul () nd (5). The reidul tree nd trin in the re outide the ellipe with the emi-xe c nd d, ccording to formule (6), (5) nd lo formul (), (5) will be: / y + +, G ( + b) G ( b) Gb G b = co + y ; yz = in x ; (9) b = y+ b co + y. / b yz = x x + + yz = x+ x. Gb Gb In the re inide the ellipe with the emi-xe c nd d ccording to formul (), (5), (5) nd lo formul / (), (), (5), we hve: x + +. / Gb G b (5) b = G y G y + + G y, Gb Gb In the conidered re, the formul of reidul / replcement my be written in the imilr wy. Formul b yz = G x G x + + G x. () (5) re vlid in the ce when condition () nd Gb Gb () re fulfilled, i.e. if t totl unloding there pper / econdry pltic deformtion. But if one of the b condition () nd (), i not fulfilled, then t removing = y y + + Gb y. Gb the torque there pper eltic unloding. Herewith, the reidul tree nd trin my be clculted by A.A. / b Ilyuhin eltic unloding theorem [5]. According to yz = x x + + x. Gb Gb thi theorem, the reidul ought-for vlue re determined by formul (7). Herewith, the quntitie Ftigue Filure of Ovl Br t Pulting Torion: ow, yz,, yz, u x, u y, u z, re determined from () let conider ftigue filure of the conidered br under - (6), () (), (), (9), (), (). pulting torion. We will ue cyclic dmge nd the, yz,, yz, ux, uy, uz,. But the quntitie re cyclic durbility condition. According to [5], the cyclic the tree, trin diplcement nd twit exiting in dmge condition h the following form: yz (5) (5) (5)

7 ( t ( )) dk = ( ( )) ( ( )) t t ( t ( k) ) ( t ( k) ) ( t ( )) dk = ( ( )) ( ( )) t t ( t ( )) ( t ( )) k k ( ) / t = + yz. (55) (5) In reltion (5) nd (55) t i the intenity of reidul In the ovl br the ection x = i dngerou for deformtion in the -the loding circle; i the number filure. From phyicl reoning' it follow tht filure of of loding cycle t which in the mteril the dmge the ovl br begin t the point (, - b); (,-(-b). ccumultion proce begin for t = t ( k) ; i the Furthermore, exitence of econdry pltic deformtion number of loding cycle to filure for t totl unloding ccelerte the metl filure proce. t t ( k) ; ( t ) nd ( ) t re of the Proceeding from thi fct, by clculting the intenity function experimentlly defined for ech mteril, of reidul deformtion, we ue formul (). For x =, herewith nd re the loding cycle before y = - b, we hve: ppernce of dmge in the experimentl mple to filure. In reltion (5) nd (55), nd re the ought = ( + b) ; yz =. for quntitie. The function nd my be pproximted in the form: Tking into ccount the lt reltion in (5), we get: ( t ) ; ( t ) = A = A = =,5; A =,9 ; A =, World Eng. & Appl. Sci. J., 6 (): 7-, 5 Since the br mteril i idelly pltic, then. the reidul deformtion for ny cycle of totl unloding will be the me in the firt totl (5) unloding. Thi concluion i vlid lo in the ce if in the firt totl unloding there rie econdry pltic The cyclic durbility condition i written follow: deformtion. We clculte the intenity of reidul deformtion by the formul: (56) ( ) t = + b When proceing experimentl dt of the pper [], (59) for the teel of mrk 5 the following vlue were obtined: Formul (59) hold in ny -th cycle of the pulting torion, i.e. the quntity t i independent of the number of pulting torion. Herewith from (57) it follow tht the quntitie nd coincide with the quntitie nd B, repectively. Conequently, we hve: Herewith / = B =,.5 In [5] it w mentioned tht the rtio / = B = cont hold for everl mteril. Proceeding from thi fct, condition (5) nd (55) re trnformed to the form: = BA ( + b) ; = A ( + b). dk dk = ; = B. ( t ( k) ) ( t ( k) ) (57) At the bove mentioned vlue, B, A for the teel of mrk 5 nd lo t the vlue M G Ue condition (57) for determining the mount of =, =,5 pulting torion cycle to the firt dmge nd filure of the conidered ovl cro ection br. To thi end, we nd 6,5 define the intenity of reidul deformtion (k), where b = for the quntity nd we get the t i the current mount of torion. following vlue: =,. cycle, =,67. cycle. 5 5

8 World Eng. & Appl. Sci. J., 6 (): 7-, 5 COCLUSIO REFERECES Condition of ppernce of econdry pltic. Sokolovkii, V.V., 969. Theory of plticity Mocow: deformtion t totl unloding fter preliminry eltico- Vyhhy Schkol, pp: 6. pltic torion of ovl cro ection br were determined.. Ilyuhin, A.A., 9. Plticity, Prt I Eltico-pltic The reidul tree nd deformtion t eltic deformtion - Mocow: Gotechizdt, pp: 76. unloding pltic deformtion, tht re preerved in the. Mokvitin, V.V., 965. Plticity t vrible loding ovl br t elctico -pltic torion, re determined. Mocow: MGU, Publ., (Ruin), pp: 6. The number of the cycle of pulting eltico-. Ivnov, V.S., Yu I. Rgozin nd.a. Vorobyev, pltic torion t which the firt dmge pper nd 967. Regulritie of metl filure t ttic nd cyclic filure begin were determined. lod. In the book: Termopltichnot mterilov: kontruktivnikh element Iue IV, Kiev, ukov Dumk, pp: Tlybly, L., 995. Kh To mll cycle nd therml ftigue, Proc. of the I Republicn Conference on mechnic nd mthemtic devoted to 5 yer of AS of Azerbijn, Bku, June, 965, Prt I. Mechnic, Bku, pp: 9-96.