 Applied Mathematics, 2011, 2, 579-585 doi:10.4236/am.2011.25077 Published Online May 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Stability Loss of Rotating Elastoplastic Discs of the Specific Form Dmitrii Makarovich Lila1, Anatolii Andreevich Martynyuk2 1Cherkasy National Bohdan Khmelnytsky University, Cherkasy, Ukraine 2Stability of Processes Department, S. P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine E-mail: dim_l@ukr.net Received February 17, 2011; revised March 25, 2011; accepted March 28, 2011 Abstract A method of calculating a possible stability loss by a rotating circular annular disc of variable thickness is suggested within the theory of perfect plasticity with the help of small parameter method. A characteristic equation for a critical radius of a plastic zone is obtained as a first approximation. The formula for the critical angular velocity, determining the stability loss of the disc according to the self-balanced form, is derived. The method using which we can take into account the disc’s geometry and loading parameters is also speci-fied. The efficiency of the proposed method is shown in Section 5 while considering an illustrative example. The values of critical angular velocity of rotating are found numerically for different parameters of the disc. Keywords: Axisymmetric Elastoplastic Problem, Boundary Shape Perturbation Method, Rotating Circular Annular Disc, Stepped Disc, Stability Loss, Critical Angular Velocity 1. Introduction The analytical methods of studying the stability loss [1-6] at radial tension are known to be applied to plane discs (with constant thickness) in elastoplastic state. In  a method of calculation of possible stability loss was pro-posed for the case of the simplest non-planar rotating circular disc, namely, the stepped disc, loaded by radial stress on the boundary. This method underlies the present approach to approximate calculation of critical radius of the plastic zone and critical angular velocity of the rotat-ing annular disc of variable thickness. Besides, the real profile is roughly replaced by a step-like one, so that the disc is considered to be composed of partial annular discs of constant thickness. The applicability of the algorithm to the analysis of the small perturbations dynamics in case of the discs with arbitrary profiles is discussed. 2. Problem Statement Consider a stability loss of the rotating annular disc with an arbitrary smooth profile yr (Figure 1) as a result of its attaining an equilibrium form, different from a cir-cular one, in the plane of rotation. We will assume the disc to be almost circular, and present the equation of external boundary in its middle plane 0y=const,, being a plane of symmetry of the disc, with the accuracy to the first-order infinitesimals, in the following form =cos,2,rbd nθnd or =1 δcosρnθ (1) Figure 1. Stability loss of the disc according to self-balanc-ing form. D. М. LILA ET AL. 580 where b is the external radius of the unperturbed disc (the radius of circumference profile), =r b is the non-dimensional current radius,  is a small parameter, , n is a polar angle. Let a be the internal radius of the disc, s be the yield strength of the material, be the modulus of elasticity, E be the density, v be Poisson’s coefficient,  be the angular velocity of ro-tation and 0 be the current radius of the plastic zone for the unperturbed disc. rLet’s assume that the maximal thickness of the disc is small as compared to its other dimensions. Based on this assumption, the stresses located on the internal and ex-ternal boundaries of the disc will be considered as re-sulted from certain efforts 0i and [7,8], acting on the disc in its middle plane. =iipp p0=eepp peFor the boundary form, described by (1), we need to obtain (as a first approximation) the characteristic equation for the critical radius of the plastic zone 0 and to find the corresponding critical angular rotation velocity r. 3. The Unperturbed Elastoplastic State of the Rotating Disc Consider the equation of quasi-static equilibrium  21d =drrrr yyrr b,r (2) where 22=.b Basing on yield condition (of maximum shear theory) and taking into account that the problem statement gives =,rr iap in the plastic region 0,rar, we present the solution of linear differential Equation (2) 2d11d =ddrrrryrrryr rb  in general form =;,rri .xrap (3) Moreover, =.s (4) Taking into account the condition on the external boundary =rr ebp and yield condition (constant stress intensity), suppose that in the disc elastic region 0,rrb the stress com-ponents are =,;,rr ezrCbp, (5) =,;,ewrCbpHere the constant C is to be found. Having in mind that non-dimensional values will be used in further calculations we refer the values with the dimension of pressure to the yield strength s. The values with the dimension of length will be referred to the characteristic length b. Introducing the notations 00:=, :=,rb ab we use the continuity condition for the stress components at transition through the boundary 0=. Equating the right-hand sides of (3) and (5), and those of (4) and (6) at 0=, we get the system of equations 00;,= ,;1,,is esxp zCp  01=, ;1,.eswCp Its solution 00=,=sC  fully describes the stress state (3)-(6) and determines the dependence of the angular velocity of disc rotation on the radius of plastic region. 4. Principal Result Along with relations (3), (5), (6), consider an approxi-mated stress state, obtained at dividing the given disc of an arbitrary profile into partial discs of constant thick-ness 10 (Figure 2). In  it has been shown that the dependences corresponding to (3), (5), (6) for the stepped annular disc are 2, ,2nhh211221202101,,31,,=31,3spsjjsCCC,,,, (7) 221, 2,0221, 12, 110221, 2,100 00,,,,,=,,jj jjj jjenn nnCCCCCC  ,, (8) Figure 2. The disc of arbitrary profile divided into partial discs of constant thicknesses. . (6) Copyright © 2011 SciRes. AM D. М. LILA ET AL.581 ,,   221, 2,0221, 12, 110221, 2,100 00,,,,,=,,jj jjj jjenn nnCCCCCC  (9) where 111100=,,=nnrbrb, 0=1n, =38s , =318s, and the constants 1,,jCC and  1, 2,2,0,,,1,,0jjnCnCCC are found as solutions of the systems 21=1 ,3isspC  22111 2111223222322212211 1111=1,331=133 1=133ssssjjjj jjsj sjCChhCChhCChh       2, and 1, 2,1, 2,1, 12, 1111, 12, 1111, 2,0001, 2,000=,=,=,=, =,=jjjjjjjjjjjjjjjjnnnnnnCCxsCCxtCCxsCCxtCCsCC t respectively. Here 22100=1, ,=1jjn nxx1, 2200 0000021 2020 000=()8=,311=,31 24δ31=,33 3seinn nnssnnnsqppRAR BSASBQAQ BDqbQd f    101 0103 0=δ,=δ1,RdSd 21020 000101 0103 00000020003112 11=131100=131 24δ31=,33 3=δ,=δ1,1=,=,=,221δ=,δ=(),31δ=( ),=0,=,jjjkkkkjjjkkkkjQf dRfSfxx xxdfxxxhhhhhhh hh2    whereas 01 0100,, ,,,,jjnnnnststst  and 0,nA0,nB0nD, are found from the recurrence relations  111 1111 1222 2222 2000 000=,= ,=,=,=,=, =,=iijjssjjjjjjjjjjjjjjjjjjjjjjjjnnjnjnnnppsQRStQ RSsAsBtCtAsBtCsAsBtCtAsBtCsAsBtCt As     00,jnj nBt C where 1111112111111121111 1112111111=,=,=,==, =,=,=,=jjjjjjjjjjjjj jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjAdafcB fCdbfbAfadcBdCfbdbAdaAf cAABdaBf cBBCdaCbfcC,  11 1211111121111112111111111100,=,=,=, =jjjjjjjjjjjjjjjjjjjjjjjjjjjjnnCbAfaAdcAABfaBdcBBCfaCbdcCCbAda 11001111000011 1000011110000,=,nnnnnnnnnnnnnnAfcAABdaBfcBB   11 Copyright © 2011 SciRes. AM D. М. LILA ET AL. Copyright © 2011 SciRes. AM 582 11110000 0111110000011 111 110000000011 111 1100000 000111100000111000=,=,=,=nnnn nnnnnnnnnnnnnnnnnnn nnnnnnnnnnnnCdaCbfcCCbAfaAdcAABfaBd cBBCfaCbdcCC     110001111 11111101111111,=,=1,,,=,=, =2,,,=,=,=,=,2=,2nkkjjjjjjjj jjjjjjkkkkkkkkkk kkkkkkkkkkkbCDkj nDgdfDdaD gfcDDgjj nhhhabhhxhh xxcdhxxxfgx   1101=,=, ,1.kkkkhhhxkj n Dependences (7-9) with account of the relation provide a zeroth approximation to the solution of the problem on plastic equilibrium, determining the position of elastoplastic boundary. In addition, 0=1p 0100 0000 00021d1=d=3,=11=3,eienn nnssnn neeAppAQBQARB RAS BSCAA      where 0120012003120000=83=,830< 1,0<1,1=,0,>0,>1,=1,=isiieininnieeeinnnes eipSASBRAR BQAQ BDp  n for 0i, and 3100000310010813== 8113=eeiennnesennis ieRARBSAS BpRAR Bp   ,n 40,=1ij ija, for . 0e is the determinant of the matrix The number of sections 0n of the stepped profile, which approximately substitutes a real one, still remains unknown, and constant half-thicknesses of partial annu-lar discs can be introduced by the average theorem: 0100=d,=,1,,rjjrjjnhyrrbabara jjnn. (11) where (10) where 0000000011 11=1,1 ,InsAdE12 1 113 1114 1121 2122 2123 2124 21311,1322,1=,1,=,1,=,1,=,1,=,1,=1,1 ,=,1,=,=,II n sIII nsIV nsInsII nsIII nsIV nsjjaaAd EaAdEaAd EanAd EanAd EanAd EanAd Eaq aq First assume that 0 equals to a certain fixed small natural number. Then, with regard to (10), one has a characteristic equation [2-4,7] n0=0, D. М. LILA ET AL.583 ,,,are the known functions of two vari-333,1344, 1415, 1426,1437,1448, 1=,==,==,=jjjjjjaq aqaq aqaq aq  ,,IIVdd ables, and 0nA, , and 0nB0nC1, 18,1,,jjqqoreover, the critical are tions. Mfound fromangular velocity recurrence rela, co 0rrondingdius esp to the critical raof the plastic zone, 0,1nding ins to be seen  depea, is obtained fron the type of cwhether om on-the known formtour load ip, pulae . It rem,e0 and  are exacvalues for the dit apprwith goximaven prtions oofile f the corresponding sc iyr. Let  be an arbitrary positive number. Let it be con-nected with the absolute error of the stress state, ap-peared due to transition to a stepped disc, by the condi-tion 00,,10max sup pe ex,100 0, ,sup sup,zw      (12) where the functions  (3nd (7)-(9) are taken for 00=), (5), (6) a. If for 0 being the solution of charac-teristic equion11), inequality (12) fails, one should take 00:= 1n, redetermine atn (jh, and also (7)-(9) ac-cording to (10) and solve Equation (1ce1) on again. The fulfillment of condition (12) with new 0 allows to complet solution of the problem on the stability loss of the disc with given profile, with the accuracy of . If equality (12) peated with fails, the descri1bed d so onprocedure uan. 5. Example Let’s calculate the stability loss for the disc of a hymst be re-00nn:= per-bolic profile =,,>0.syk ks (13) Many real profiles can be approximately expressed by Equation (13). For such discs, as well as for those of constant thickness (=0s in (13)), the stress-strain state can be obtained in a closed form . From Equality (4), Equation (2) in the plastic region is presented as r2d1=srr.drr srrr rb (14) f the corresponding initial problem is of Solution (3) othe following form  2=21 1221133sssrr iara psssbor srsb  211;,=13isspxss 121.13ssissss11p  (15) In the elastic region the stress components of the un-perturbed annular disc with a hyperbolic profile can be sought as  21212 212121222=,=11,rr CrC rrbsCrsC rrb   (16) where 1C, 2C, 1, 2, ,  ndiare yet to be speci-fied. Substitutionng expressions (16) for of the corresporr and  in equilibrium Equation (14) gives 1=.3s (17) After substitution of expressions (16) into 1dd1=0,=ddrrrrrmr mmrr , obtained by exclusion of radial displacement coupling equations for deformations and stresses, based on (17), we get from the  31 3=,=.831831msm msmmm (18) Busihe method of undetermined coeffiesides, ng t-cients, find the indices 12,: 21,2 =11 .24sssm (19) The condition on the external boundary leads to the relations 1212=,=.eCbpCbC C Taking them into account in (16), from the system of stress continuity equations at transition through the elas-toplastic boundary, we get 1201 011=espsC 212010,11ssss  (20) Copyright © 2011 SciRes. AM D. М. LILA ET AL. 584  1112211101 0210 120131020 10=11141113ssssisspsssss12 121200210()11epss 22220 20143sss,    (21)    122,;1,=,eessssppzC CC (22) 12122,;1,=11,eessssppwCs CsC     (23) where  1212112122112 120011210 0102 0121 101 01202102220 101=1111(1)( 1)(1)1,=4 43iiis2sseissipssssssss       12213120 020101311,=sseieissssspp    for , and >0i123120011=11111eeessiepsss  1212121010 20311101 0120210111111,=ssseieiesssssspp      for ze the dynamics of small perturbations let’s first calculate half-thicknesses >0e. To analyjh. In terms of (10), we have: 110000111=,111, ,.ssjskn jjnnhbsjn(24) hen wrmine dependences (7)-(9) (for as yet unknown Te dete0) and characteristic Equation (11) itself. Its solution 0 allows proceeding to the verification of estimatio with previously given n (12). In someases the exact upper limits in inequality (12) can be found utions of thobal extremum problem for continuously differentiable functions (at discontin points canalytically as the soluitye gl011,,n he values of it is necessary to use one-sided lits as tfunction: the right-side limit at mi1j and thone at e left-side j of each seg- ment 1,,jj 1  )thod appears to be easier and . However, the numerical m- more versatile tool to verify condition (12). It is reduced to finding the maximum of the set of limited nonnegative piece-wise continuous functions, given at e00=), in the poi by using relations (15), , (23) a (7)-(9nts of quite dense discre-tizolic disc with and depending upon . Here (22) ndation of the corresponding segment. Table 1 gives the results of problem solution for a hyperb =0.005k, =1b, =2s0n=2n==0ab .2, =0.3, =0.01sE, =0i, =0e, =1 3, ei 6. Concluding Remarks The proposed scheme allows determining the critical =0. Table 1. Critical radius and squared relative critical veloc-ity. n0 3 10 20 25 30 β0* 0.73310.8399 0.9199 0.93590.946622q 0.88635195 0.50420.49390.5929 0.Copyright © 2011 SciRes. AM D. М. LILA ET AL. Copyright © 2011 SciRes. AM 585metudying the discs,s c annular aisrbitr prof . Russian, Vyshcha shk, Kiev, 1989. ing Discs, Close to Circular Ones,” Izvestiya Akademii Nauk SSSR, Otdelenie Tekhnicheskikh Nauk, in Rus-,No. 1, 7, pp. 141-144.  D. Ivl.shtuMnTheorc Boin Rusan, NaukaMo ow,lastoviscoplastic Ro- .07.008 t II—Burst of a Superalloy Turbine On the Instability of Rotating Elastoplastic lastic Annular Disc,” In-. irger, “Stress Calculation stroyeniye, Mos- ickness,” Applied Mathematics, Vol. 1, No. 5,  M. Mazière, J. Besson, S. Forest, B. Tanguy, H. Chalons and F. Vogel, “Overspeed Burst of Eradius and critical angular velocity of rotating disc with the given profile for known load paraters. This en-tating Disks—Part I: Analytical and Numerical Stability Analyses,” European Journal of Mechanics—A/Solids, Vol. 28, No. 1, 2009, pp. 36-44. doi:10.1016/j.euromechsol.2008ables s whose unperturbed elastoplastic state can be obtained in a closed form . Except the discs of constant thickness and the hyperbolic discs, conical, exponential and equal resistance discs , as well as compound dics of the mentioned profiles are referred to this type. Besides, neglecting the procedure of verification of condition (12) due to sufficient increase of th M. Mazière, J. Besson, S. Forest, B. Tanguy, H. Chalons and F. Vogel, “Overspeed Burst of Elastoviscoplastic Ro- tating Disks: ParDisk,” European Journal of Mechanics—A/Solids, Vol. 28, No. 3, 2009, pp. 428-432. doi:10.1016/j.euromechsol.2008.10.002  D. M. Lila, “e number of stepped disc sections 0n, we get a method of calculating the possible stability loss (including the eccentric case) for fast rotating elastoplastind solid dcs of an aaryileStepped Annular Disc, Loaded over the Boundary in the Middle Plane,” International Applied Mechanics (in Press).  D. M. Lila and А. A. Martynyuk, “On the Development of Instability of Rotating Elastopternational Applied Mechanics (in Press) 7. References  А. N. Guz and Yu. N. Nemish, “Method of Boundary Form Perturbation in the Mechanics of Continua,” in  D. D. Ivlev, “On the Loss of Bearing Capacity of Rotat- sian 195D. the ev and Ly of Elas V. Yertoplastiov, “Perdy,” rbation siethod i, co K. B. Bitseno and R. Grammel, “Technical Dynamics,” Gosudarstvennoe Izdatelstvo Tekhniko-Teoreti- cheskoy Literatury, in Russian, Vol. 2, Moscow and Leningrad, 1952.  I. V. Demianushko and I. A. Bof Rotating Discs,” in Russian, Mashinow, 1978.  A. M. Zenkour and D. S. Mashat, “Analytical and Nu-merical Solutions for a Rotating Annular Disk of Vari-able Thsc 1978.  L. V. Yershov and D. D. Ivlev, “On the Stability Loss of Rotating Discs,” Izvestiya Akademii Nauk SSSR, Otdele-nie Tekhnicheskikh Nauk, in Russian, No. 1, 1958, pp. 124-125. 2010, pp. 431-438.