Determination of the rheological properties of thin plate under transient vibration

Transcription

1 (3) Deenaon of he heologcal popees of hn plae unde ansen vbaon Absac The acle deals wh syseac analyss of he ansen vbaon of ecangula vscoelasc ohoopc hn D plae. The analyss s focused on specfc defoaon odels of plae and fo specfc lnea odels of heologcal popees. The vscoelasc soopc and ansoopc (ohoopc) aeals wee nvesgaed. The plae s loaded by geneal ansen pulse. The e and coodnae dependences of he fundaenal quanes dsplaceens, oaons, sess and defoaons n abay pons of plae unde ansen vbaon, e. g. he analyss of sess and defoaon waves n plaes wee nvesgaed. The seleced odels ae defned by consuve equaons fo sess and defoaon dependence. The soopc and ohoopc odel s consdeed. The analyss esuls depend on qualy of he deened echancal popees of he aeals. The sandad ehods coonly used ae e-consung and no accuae enough. The new ehodology of deenng he aeal paaees decly fo nvesgaed plae s poposed and poven. Keywods Rheologc popees, hn plae, ansen vbaon Jan Skočlas, Blanka Skočlasová, Josef Soukup Jan Skočlas, M.Sc., Ph.D.: Czech Techncal Unvesy n Pague, Faculy of Mechancal ngneeng, Depaen of Pocess ngneeng, Techncká 4, 66 7 Paha, Czech Republc (Jan.Skoclas@fs.cvu.cz). Blanka Skočlasová, M.SC. (Ph.D. suden): Jan vangelsa Pukyně Unvesy n Úsí nad Labe, Faculy of Poducon Technology and Manageen, Depaen of Machnes and Mechancs, Na Oka, 4 Úsí nad Labe, Czech Republc (skoclasova@fv.uep.cz) Josef Soukup, assoc. pof., ng., CSc.: Jan vangelsa Pukyně Unvesy n Úsí nad Labe, Faculy of Poducon Technology and Manageen, Depaen of Machnes and Mechancs, Na Oka, 4 Úsí nad Labe, Czech Republc (soukup@fv.uep.cz). INTRODUCTION The heoy of he plae vbaon belongs o he os exploed eal of echancs of defoable plaes. Auhos have deal wh hs eal oe han yeas. The hunded sudes and onogaphs have been publshed ll pesen e. The esuls of vbaon nvesgaon wee obaned by any analycal, nuecal and expeenal ehods. The os sophscaed analycal soluon s focused on ansen vbaons of hn elasc soopc plae. The ansoopc plae e.g. ohoopc plae s less coonly dscussed, e.g. []. The nvesgaon of ansen vbaon of vscoelasc plae appeas n he fs half of he las cenuy and hs poble has been exaned by sgnfcanly less acles [7]. The essenal evoluon of ansen vbaon of he vscoelasc plae begun afe echnology developen of polyes (acoolecula aeals) [6] and coposes unl second half of las cenuy, fs fo Kchhoff s odel of plae [3, 4], lae fo Raylegh Lan Aecan Jounal of Solds and Sucues (3) 89 95

2 9 Jan Skočlas, Blanka Skočlasová and Josef Soukup / Deenaon of he heologcal popees of hn plae unde ansen vbaon odel. The nvesgaon of he vscoelasc ansoopc plae s pogessng eal of echancs now. The acle deals wh he nvesgaon of ansen vbaons and sess wave popagaon n vscoelasc ohoopc hn plae. The phenoenologcal odels ae coonly appled fo descpon of aeal echancal popees unde ansen saes of vscoelasc bodes. The consuve equaons defne elaons of sess and defoaon of nvesgaon connuu by supeposon of elasc and vscoelasc coponens (epesened by spng and vscous dape especvely). The dffeen odels of vscoelasc heedy aeal wee developed by vaous cobnaons (paallel o seal) of hese eleens, e.g. Vogh-Kelvn, Maxwell, Zene ec [5]. The geneal poble s deenaon of he sess odulus of he vscoelasc ohoopc aeal (e.g. copose). The conepoay ehods of sess odulus deenaon ae based on sac easueen of he odulus of ndvdual eleens. The fnal odulus of odel s deened fo conveson of easued odulus by volue facons. Ths ehod s no sasfacoy accuae and he esuls obaned fo dynac loadng of he odel ae dspuable. These aeals ae wde used, e.g. n auoove ndusy, al vehcles, planes, shps, space eseach, cvl engneeng ec. The coposes and lanaed aeals have ndspuable pefeences (lowe wegh, hghe sengh, e sably ec.). I s obvous ha he hgh equeens wll be deanded o copuaonal ehods fo hese aeals applcaons n he fuue. The pesened conbuon deals wh one of he poan assupons fo accuacy poveen of hese calculaons and deenaon of elascy odulus of hese aeals unde dynac loadng. The poposed ehod would be applcable fo copose and sandwch consucon and lanaes. SOLUTION Mos of he coonly used woks fo nvesgaon of sess analyss n lnea vscoelasc bodes solve consuve equaons n dffeenal fo. The feaues of hese lnea odels fo one axs enson ae descbed n Alfey by equaon n p p q q whee p, q ae aeal paaees, - defoaon, - sess, n nube of aeal paaees p, nube of aeal paaees q. If = n hen s val case q q q... cons. p p p (a) When n, he poan cases ae fo = (n ) o = (n+). These odels coespond o fundaenal phenoenologcal odels of vscous connuu. I s necessay o expeenally Lan Aecan Jounal of Solds and Sucues (3) 89 95

3 Jan Skočlas, Blanka Skočlasová and Josef Soukup / Deenaon of he heologcal popees of hn plae unde ansen vbaon 9 oban he values of (+n) aeal paaees, when hese odels ae appled. The dffeenal equaons of consuve equaons ae oe coplcaed unde ul-axs enson. Fo wo-axs enson (noal sess x, y, n axs decon, and shea sess ), he consuve equaons fo heeday odel of vscoelasc aeal could be expessed by equaon ha has analogous fo o Alfey s fo fo one-axs enson. Noal sess n,,,,,, p p q q q q fo, {x, y} and. Shea sess n,,,, p p q q (3) The consuve equaon could be expessed n fo by echncal paaees (consdeed basc sples odels). Vog-Kelvn odel Noal sess fo soopc aeal y (4a) Noal sess fo ohoopc aeal ( ) (4b) Shea sess G (4c) whee, G ae he Young and Kchhoff odules, λ, ae he coeffcens of noal and angenal (shea) vscosy,, ν ae he elasc and vscous Posson aos. Lan Aecan Jounal of Solds and Sucues (3) 89 95

4 9 Jan Skočlas, Blanka Skočlasová and Josef Soukup / Deenaon of he heologcal popees of hn plae unde ansen vbaon Maxwell odel Noal sess fo soopc aeal (5a) Shea sess Noal sess fo ohoopc aeal (5b) G, whee,. (5c) I s possble o oban he equaons fo sess coponens n negal fo by soluon of dffeenal equaon (5a) fo, o (5c) fo especvely. Noal sess fo soopc aeal ( ) e d (6a) Afe negaon by pas, we go e d (6b) Noal sess fo ohoopc aeal ( ) e d (6c) Afe negaon by pas ( e ) d (6d) Shea sess Lan Aecan Jounal of Solds and Sucues (3) 89 95

5 Jan Skočlas, Blanka Skočlasová and Josef Soukup / Deenaon of he heologcal popees of hn plae unde ansen vbaon 93 ( ) G e d, whee G (6e) Zene odel Noal sess fo soopc aeal (7a) Noal sess fo ohoopc aeal ( ) (7b) Shea sess G G G G (7c) We ge he equaons fo sess coponens n negal fo by soluon of dffeenal equaon (7b) fo, fo ohoopc aeal and zeo nal condons. ( ) e d (7d) The specfc aeal paaees, G,,, ae necessay o deene by expeens. Moe ofen s deened fo sple bodes wh sac expeens and hen s appled o bodes wh oe coplcaed shapes and coplcaed loadng pocess. The nube of paaees ases fo pacula cases enoned above. The elaonshp beween sess and defoaon fo heeday vscoelasc aeals s possble o expess by Duhael negal. Inegal opeao fo one-axs enson expesses he fundaenal consuve equaon. R R( ) d, especve J J( ) d (8) Lan Aecan Jounal of Solds and Sucues (3) 89 95

6 94 Jan Skočlas, Blanka Skočlasová and Josef Soukup / Deenaon of he heologcal popees of hn plae unde ansen vbaon The elaonshp of boh expessons n he dffeenal (Alfey) and negal (Volea) fo s possble o foulae afe Laplace ansfoaon (s paaee of ansfoaon) Rs n p s s q s (9a) Js n q s s p s (9b) whee R(s).J(s) = s -, and R(- ).J( ) =. The Bolzann pncple of supeposon allows o we by adusen of fo p = subsung no (8) q K( ) d The elaonshp of boh soluons s obvous fo copason of (5a), (7a), (8) and. Concluson I s oe cofoable and less e-consung o use consuve equaons n poposed fo (appoxaon of e funcon, elaxaon odule K( - )), han o deene of any paaees. Fo equaon ase he necessy of deenaon of paaee q and funcon K=Ae -, whch appoxae funcon of vscous coponen elaxaon (odule), paaees A and. The appoxaon of hs funcon s less dffcul han deenaon of aeal paaees (, G,,, ). In case of ansen saes nvesgaon fo hn vscoelasc plae vbaon s possble o deve equaons fo sess coponens o dsplaceen (e dependen) by negal expesson of consuve equaons (8) and especvely. The equed paaees could be obaned fo copason of sess o dsplaceen coponens fnd ou by expeenal and analycal ehods. The elaxaon odule funcon could be expeenally deened decly on nvesgaed body. The ehodology fo expeenal nvesgaon of hs funcon was poposed and s unde paen applcaon. Lan Aecan Jounal of Solds and Sucues (3) 89 95

7 Jan Skočlas, Blanka Skočlasová and Josef Soukup / Deenaon of he heologcal popees of hn plae unde ansen vbaon 95 Acknowledgeen Ths wok has been suppoed by gan IGA Inenal Agency of Unvesy of Jan vangelsa Pukyně n Úsí nad Labe and gand GA ČR, nube of poec /7/946. Refeences [] Alfey, T.: Non-hoogeneous sesses n vsco-elasc eda. Q. Appl. Mah., pp. 3-9, 944 [] Ghugal, Y., M., Sayyad, A., S.: Fee vbaon of hck ohoopc plaes usng gonoec shea defoaon heoy. pp. 9-44, Vol. 8, No. 3/9, Lan Aecan Jounal of Solds and Sucues. [3] Kolsky, H.: The popagan of sess waves n vscoelasc solds. J. Appl. Mechancs Reves, Vol., No 9, pp , 958. [4] Lean, M., J.: The lnea heoy of vscoelascy. ncyklopeda of Physcs, ed. S. Flügge, Vol. Va/3 Mechancs of solds III., ed. C. Tusdell, Spnge, Beln, 973. [5] Macosko, Ch., W.: Rheology. Pncples, Measueens, and Applcaons. Wley-VCH, New Yok, 994. [6] Rande,., e al: Polye Vscoelascy. Macel Dekke, N.Y.- Basel, [7] Tschoegel, N., W.: The Phenoenologcal Tudoy of Lnea Vscoelasc Behavo. Spnge, Beln, 989 Lan Aecan Jounal of Solds and Sucues (3) 87 93