Available online Journal of Scientific and Engineering Research, 2018, 5(10): Research Article

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1 vilble online Jounl of Scienific n Engineeing Resech, 8, 5():5-58 Resech icle ISSN: 94-6 CODEN(US): JSERBR Soluion of he Poblem of Sess-Sin Se of Physiclly Non-Line Heeiily Plsic Infinie Ple wih Hole he cion of Inenl Pessue Mmmov M.., Mmmov H.. Insiue of Mhemics n Mechnics of he Nionl cemy of Sciences of ebijn. 9 Bkhiy Vhbeh see, Bku, ebijn, Z 4 Mmmov Mehibn li kii is oco of hysic-mhemicl sciences, ocen, Leing Reseche he Ceeing heoy emen of Insiue of Mhemics n Mechnics of he Nionl cemy of Sciences of ebijn Mmmov Hijn li kii is Resech ssocie he Ceeing heoy emen of Insiue of Mhemics n Mechnics of he Nionl cemy of Sciences of ebijn bsc he sess-sin ses of hysiclly non-line heeiily lsic infinie le wih he hole he cion of inenl essue e eemine. he known non-line eemining elions of V.V. Moskviin e use. he oblem is solve by meho of successive oximions. he exc nlyicl soluions of oblems of ech oximion e foun. I is inouce he gumens in fvou of using convegence of consiee oximions. Keywos sess-sin se, efomion, visco-elsiciy oblem, ceeing Inoucion Consie he infinie le wih he cicul hole of ius, which is une he cion of he unifom essue lie o he conou of he hole n i hs fiel of homogeneous n no eening on he ime emeue. he meil of le is ne o he mechniclly incomessible n hs hysiclly nonline heeiily elsic oeies []. We lso noe h he known missions, which e ccee in le comuions in limis of elsiciy, emin vli in ou cse oo. Consequenly in he consiee le he lne sess-se is elie. Use he cylinicl sysem of he cooines,,. his we hve: Hee ; ; ; ; ; ; ; ; ; ; ;. ij n i, j, ij, e he comonens of sesses n efomions ensos esecively. s eeminive elions of hysiclly non-line sequenil elsiciy we shll use he V.V. Moskviin s [] known elions: eij f sij f sij, () Jounl of Scienific n Engineeing Resech 5

2 Mmmov M & Mmmov H Jounl of Scienific n Engineeing Resech, 8, 5():5-58 Hee is momeny elsiciy moule; is men efomion, coefficien of line exnsion; sess, s ij sij eij ij is Konecke symbols; sij. () ij ij n ij efomion evio, ij ij - is fcionl viion of cciy; is ij ij evio of he sesses of ijij ij, is men is inensiy of sesses; is funcion of heeiy of kenel ceeing; f is funcion of hysiclly non-lineiy. In confomiy o ou oblem ; of he sess Besies, s e we hve ; e ; e.. Fo he inensiy. () ; s ; lying he ls elions in () we shll ge he following wo ineenen equions: s f f, (4) f f 6. (5) Fo he semen of he oblem we shoul he elion () o he elions (4), (5) which we wie in he fom (6) he iffeenil equion of he equilibium comibiliy coniion n he bouny coniions We lso noe h he emuion,, ;. (9) n u is connece wih he efomions by Cuchy elions ; (7) (8) Jounl of Scienific n Engineeing Resech 54

3 Mmmov M & Mmmov H Jounl of Scienific n Engineeing Resech, 8, 5():5-58 u u,. () We shll solve he oblem (4)-(9) by he successive oximions meho woke ou in []. Following [] eesen he funcion f in he fom f f f. In his cse he elions (4) n (6) ke he fom. iniil oximion we cce, (). 6 () o () n () (5)-(9) wih coesoning ue inices. he obine oblem is of hysiclly nonline visco-elsiciy oblem. In cse of line elsiciy he sess comonens n e eesene in he fom:,. () s we see he comonens of meil e no conine in he soluion () e heefoe he fomule () e lso soluions of he oblem (), (), (6-9). his n fom () follows h llowing fo (4) fom he coniions (6) we fin * whee.. (4). using () fom () we eemine *, (5) he efomion comonens n we fin wih using he elions (4), (5): * * ;. (6) Consequenly by solving he oblem on iniil oximion ll unknown vlues e foun which s i is esy o be convince sisfy ll necessy elions. Now following he fomul () clcule he vlue :. Le f be enie funcion f. Jounl of Scienific n Engineeing Resech 55

4 Mmmov M & Mmmov H Jounl of Scienific n Engineeing Resech, 8, 5():5-58 Jounl of Scienific n Engineeing Resech 56 hen f, f ; f. Hee n e he known meil consns eemine fom he exeimens. he nex oximion he elions (4) n (5) will hve he fom *, (7). (8) he oblem semen of he consieing oximion will be close wih iion o (7) n (8) he elions (6)-(9). Inouce he following noions:,, (9),,. Wih using he noions (9). he elions (7), (8), (6)-(9) un ino he elions:, (), 6 (), (), (), (4) ;. (5)

5 Mmmov M & Mmmov H Jounl of Scienific n Engineeing Resech, 8, 5():5-58 Jounl of Scienific n Engineeing Resech 57 he oblem ()-(5) is of line visco-elsiciy oblem whee ene he volume n sufce foce eemine by he chceisics of hysicl nonlineiy geomeicl chceisics n he given essue. Solving his oblem wie he nlyicl exession fo he iniil vlues: 4 4, (6) 4, (7), 4 (8), 4 (9) 8. () By immeie subsiuion we e convince he soluion (6)-() is he exc soluion of he oblem ()-(5) in cse of biy kenel of he ceeing. ccoing o he noions (9) he efomion comonens, n will be eemine by he fomule (8)-() esecively he sess comonens of n wih using (6) n (7) will be wien in he fom 4, () 4. () Hence he soluion of he oblem of consieing oximion becme known in nlyic fom. he oblems of he following oximion e consuce by he nlogous wy. If o he elions ()-(5) ly he

6 Mmmov M & Mmmov H Jounl of Scienific n Engineeing Resech, 8, 5():5-58 Llce nsfomion hen he ocess of oximions in imge will be nlogous o he ocess of oximion in he meho of elsic soluions.. Ilyushin in heoy of elsic-lsic efomions [], whose oof of convegence is known [4], [5]. heefoe close o he el bouneness on he funcion f, we cn lk bou convegence lie in ou oblem of oximions. Following fom his le be esice consiee in he ls cse of oximion n fo he soluion of iniil oblem (4)-(9) cce oximely: whee,,,, (), esecively.,,,,, xe eemine by he nlogous exessions (), (), (8), (9), Using he fis elion () n (9) we cn eemine he unique comonen of he veco emuion u. In conclusion noe h if in cse of elsic o hysiclly line heeiily elsic meil of le evey oin he se of cle shif is elie hen s show he soluion in cse of hysiclly non-line heeiy elsic meil such se of le oes no hve. Refeences []. Moskviin, V.V.: Resisnce of visco-elsic meils. М., Nuk, 7 (97) []. lybly, L.Kh.: o he quesion of efomion n he esucion of viscoelsic boies involving he emeue boy. Iv. N SSSR. Mechnik veogo el, No, 7_9 (99) []. Il yushin,..: Plsiciy. P I. М., osekhi, 76 (948) [4]. Bykov, D.L.: On some mehos of soluion of he oblems of lsiciy heoy. In book: Elsiciy n non-elsiciy. М., Iv. MU, v. 4. 9_8 (975) [5]. Voovich, I.I., Ksovskiy, Yu.P.: On meho of elsic soluions. Dokly N SSSR. v. 6, No 4, 74_74 (959) u : Jounl of Scienific n Engineeing Resech 58