Available online Journal of Scientific and Engineering Research, 2017, 4(2): Research Article

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1 Avlble onlne Jonl of Scenfc nd Engneeng Resech, 7, 4():5- Resech Acle SSN: CODEN(USA): JSERBR Exc Solons of Qselsc Poblems of Lne Theoy of Vscoelscy nd Nonlne Theoy Vscoelscy fo echnclly ncompessble Bodes ehbn A. medov, Lf Kh. Tlybly nse of hemcs nd echncs of he Nonl Acdemy of Scences of Azebjn. 9 Bkhy Vhbzdeh see, Bk, Azebjn, AZ 4. mmdov ehbn Al kz doco of physc-mhemcl scences, docen, Ledng Reseche he Ceepng Theoy depmen of nse of hemcs nd echncs of he Nonl Acdemy of Scences of Azebjn. Absc n he ppe we ce heoems h by flfllng he condon of mechncl ncompessbly of mel edce he poblem of nonlne heoy of vscoelscy wh V. V.oskvn deemnng eqons o he poblem of physclly nonlne heoy of elscy of heonomcl bodes. Unde he noed condon he epesened heoems llow o edce he poblem on lne heoy of vscoelscy o he ppope poblem of elscy heoy. The sggesed heoems e llsed on n exmple of poblems. Keywods Qselsc Poblems, Vscoelscy, Lne Theoy, Nonlne Theoy. Semen nd solon of he genel poblem. Gve he semen of qssc poblem of nonlne heoy of vscoelscy wh V.V. oskvn deemnng eqons [] fo mechnclly ncompessble bodes G e o j f sj f sjd; ; (.) sj / G ej ejd; ; (.) nd j, j F ; jl j S R ; S o; (.3) j, j j, / o j, kl kl, j k, jl jl, k. (.4) Hee, j, k, l,,3;,, e he componens of pemons, defomon nd sess, j j especvely; ej j j; sj j j; jj / 3; jj / 3; j e Konecke / / symbols; e e / 3 ; 3s s / ; modls of mel; L e mllyesolven kenels; j j F nd j j G cons f, e he fncons of nonlney of mel; R e volmec nd sfce foces, especvely; s n nsnneos she nd e bondy pemons. Jonl of Scenfc nd Engneeng Resech 5

2 medov A & Tlybly LK Jonl of Scenfc nd Engneeng Resech, 7, 4():5- The followng heoems hold. Theoem. Poblem (.), (.3), (.4) hs he followng solon. whee he qnes d; j j jd, j j, (.5) of mechnclly ncompessble bodes, j, j e he solons of he followng poblem of he heoy of nonlne elscy s ; ; G ej f j (.6) j, j F ; jl j S R ; S o o L j / ;. d; (.7), j j, j, kl kl, j k, jl jl, k (.8) Hee we dop he followng denoon e j j ; s j j j j j ; / 3; / 3; 3 ; / e e / 3 ; 3s s / /. j j j j j Theoem. Poblem (.), (.3), (.4) hs he followng solon jd, (.9) ; j j; j j L whee he qnes j, j, j e he solons of he followng poblem of he heoy of nonlne elscy of mechnclly ncompessble bodes s j / G e ; ; j j j (.) j F F d ; (.) j l j S R S R d ; ; / ;. (.) j, j j, j, kl kl, j j, kl jl, k (.3) Hee we denoe j j j e j ; s ; / 3; / 3; 3 j / e e / 3 ; 3s s / /. j j j j j j The poof of heoems nd e ced o by dec sbson of fomle (.5) nd (.9) no ppope elons. Theoems nd heoems e lso vld fo f, h holds n he cse of he heoy of lne vscoelscy. j j j ; Jonl of Scenfc nd Engneeng Resech 6

3 medov A & Tlybly LK Jonl of Scenfc nd Engneeng Resech, 7, 4():5-. Exmples. ) Pe bendng of sgh bem. Accep h x3 s n xs of bem, x nd x e pncpl cenl ne xes of coss secon whose e wll be denoed by F. ove p x n enson fbes, hen x wll be nel xs. Lel sde of he bem s fee fom exenl foces nd mss foces e bsen. Le he momens eql n sze nd ppose n sgn nd whose plne of con concdes wh he plne x x 3 be ppled on he ends of he bem. Assme h bems mel s mechnclly ncompessble nd s popees e expessed by he lws of he heoy of nonlne vscoelscy (.). And he poblem on defnon of componens of sesses j nd sns j s fomed fom elons (.), fom he fs elon of (.3), he second elon of (.4), S s ken s lel sfce. Hee R he fs bondy condon (.3), whee dd bondy condon h shold be ssfed on he ends of he bem. To hese elons we 33xdF. (.) F Now, se fomle of (.5). n hs connecon, elon (.), he fs elon of (.3), he second elon of (.4) nd he fs bondy condon (.3) e edced o coespondng elons (.6)-(.8). Condon (.) s wen n he followng fom: 33 F x df. (.) Le fo he mel of he bem non-lney fncon f A be fond expemenlly,.e. he consns A nd e known. Le s solve he poblem composed of (.6), he fs nd second elons of (.7), he second elon of (.8) nd elon (.). epesen he sess componens j n he fom / 3 3 ; 33 x wheec s sll nknown fncon, s he known consns of he mel., R C (.3) Sess componens ssfy he fs wo elons of (.7) fo F h coespond o he condons of o poblem. And ssfcon of condon (.) leds o he elon C x df C. Hence C / / F 33 x s3 s. Conseqenly,. (.4) s esy o defne 33 / 3; s / 3; 33; s s 33 / 3;, whee enso componens j by eqons (.6) 33s epesened by foml (.4). Allowng fo hese elons we fnd he sn Jonl of Scenfc nd Engneeng Resech 7

4 medov A & Tlybly LK Jonl of Scenfc nd Engneeng Resech, 7, 4():5- A j 3G x ; 33 ; 3 3. (.5) s esy o check h he fond componens j denclly ssfy sx sn compbly eqons (.8). Afe deemnon of j nd j by fomle (.5) we fnd he desed componens of sesses nd sns j h se n sgh bem mde of physclly nonlne vscoelsc mel nde pe bend: 3 3 ; 33 x ; (.6) Ax 33 d 3G (.7) 33; 3 3. (.8) Solon of (.6)-(.8) concdes wh he solon of he consdeed poblem obned n [] n nohe wy. s lso esy o vefy h componens of sess (.6) nd sn (.7) nd (.8) ssfy ll necessy eqons nd bondy condons. Theefoe, hey e exc solons of he consdeed nonlne poblem of vscoelscy nde condons of powe dependence of nonlney fncon nd mechncl ncompessbly of bem s mel. b) Plne defomon of hollow hck wlled cylnde by nenl pesse. A hollow hck wlled cylnde of nenl ds nd exenl ds b s nde he con of nenl pesse p j. echncl popees of he cylnde s mel e descbed by he eqons of non-lne vscoelscy of ncompessble mel (.). Rdl sess, pephel sess, coespondng defomons, nd pemon (plne defomon) sng n he cylnde by (.), (.3) nd (.4) ssfy he elons: R d; ; G ; p b, ; z., ; (.9) (.) (.) By foml (.9) he poblem (.9)-(.) s edced G ; ; ; 33 he followng poblem: p b p d ; ; (.) (.3) ; ; z. (.4) Jonl of Scenfc nd Engneeng Resech 8

5 medov A & Tlybly LK Jonl of Scenfc nd Engneeng Resech, 7, 4():5- Poblem (.)-(.4) s pcl cse of poblem (.)-(.3). We epesen he nonlney fncon / 3 whee n he fom of he powe fncon: o z nd ge: c c c,,, c whee, B. We se (.4) fom ncompessbly condon 3 (.5) c s sll nknown fncon. Allow fo (.5) n he fs eqon of (.) nd deemne he expesson by he fncon c. Usng he obned expesson n he eqlbm eqon (.3) nd bondy condon fo 3 / BG c we deemne he componens of p p d We se bondy condon (.3) fo c p p / 3 4BG b. (.6) b, fom (.6) fnd he nknown fncon c d Allowng fo (.7) n (.6) we deemne he qny b b p / : p d. (.7). (.8) Now, sng (.5), (.7) nd (.8), fom he eqon (.9) we fnd : Pemon b b, defomons p p d. (.9), e deemned by fomle (.5) wh egd o (.7). Conseqenly, we fond exc nlyc solon of poblem (.)-(.4). Now sng fomle (.9) we deemne he desed solon of poblem (.9)-(.) b b p ; p. (.) b Afe deemnng nd, sess z s fond on he bss elons of plne defomon of ncompessble mel: of he second foml of (.9) z b. Sn componens nd e deemned on he bss Jonl of Scenfc nd Engneeng Resech 9

6 medov A & Tlybly LK Jonl of Scenfc nd Engneeng Resech, 7, 4():5- p p d / 3 4BG b. (.) We cn fnd pemon и fom he second foml of (.): whee s epesened by he foml (.). The solon of (.)-(.) concdes wh he solon of he consdeed poblem obned n [] by nohe mehod. Alongsde wh hs we cn see by dec sbson h fomle (.)-(.) s he solon of poblem (.9)-(.). Refeences []. oskvn V.V. Ressnce of vscoelsc mels.м.: Nk, 97, 33 p. (Rssn) []. medov.a. Dely fce of bs of vscos-elsc-plsc mel on cyclc bendng// Tnscons of AS Azebjn, sees of physcl-echncl nd mhemcl scences, v.xx, N-, Bk, 999, pp Jonl of Scenfc nd Engneeng Resech