Rotating Variable-Thickness Inhomogeneous Cylinders: Part II Viscoelastic Solutions and Applications

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1 Appled Mathematcs do:10.436/am Publshed Onlne Decembe 010 ( Rotatng Vaable-Thckness Inhomogeneous Cylndes: Pat II Vscoelastc Solutons and Applcatons Abstact Asha M. Zenkou Depatment o Mathematcs Faculty o Scence Kng AbdulAzz Unvesty Jeddah Saud Saud Aaba Depatment o Mathematcs Faculty o Scence Kaelshekh Unvesty Ka el-shekh Egypt E-mal: zenkou@gmal.com Receved June 010; evsed Octobe 1 010; accepted Octobe Analytcal solutons o the otatng vaable-thckness nhomogeneous othotopc hollow cylndes unde plane stan assumpton ae developed n Pat I o ths pape. The extensons o these solutons to the vscoelastc case ae dscussed hee. The method o eectve modul and Illyushn's appoxmaton method ae used o ths pupose. The otatng be-enoced vscoelastc homogeneous sotopc hollow cylndes wth unom thckness ae obtaned as specal cases o the studed poblem. Numecal applcaton examples ae gven o the dmensonless dsplacement o and stesses n the deent cylndes. The nluences o tme consttutve paamete and elastc popetes on the stesses and dsplacement ae nvestgated. Keywods: Rotatng Vscoelastc Cylnde Othotopc Vaable Thckness and Densty 1. Intoducton In ecent yeas the subject o vscoelastcty has eceved consdeable attenton om analysts and expementalsts. The stess state o a vscoelastc hollow cylnde wth the help o ntenal pessue and tempeatue eld s analyzed n the lteatue [1]. A moded numecal method s ntoduced by Tng and Tuan [3] to study the eect o cyclc ntenal pessue on the stess and tempeatue dstbutons n a vscoelastc cylnde. Talybly [4] has nvestgated the state o stess and stan o a vscoelastc hollow cylnde astened to an elastc shell unde nonsothemal dynamc loadng. Feng et al. [5] have obtaned the soluton o nte deomatons o a vscoelastc sold cylnde subjected to extenson and toson. The themomechancal behavo o a vscoelastc nte ccula cylnde unde axal hamonc deomatons s pesented by Kanaukhov and Senchenkov [6]. The detemnaton o stess and dsplacement elds s an mpotant poblem n desgn o engneeng stuctues usng be-enoced composte mateals. The analytcal soluton o the otatng be-enoced vscoelastc cylndes becomes vey complex when the thckness along the adus o the cylnde s vaable even o smple cases. Methods o solvng quas-statc vscoelastc poblems n composte stuctues have been developed by a numbe o authos [7-9]. Allam and Appleby [10] have used the ealzaton method o elastc solutons to solve the poblem o bendng o a vscoelastc plate enoced by undectonally elastc bes. In othe wok [11] they have used the method o eectve modul to detemne the stess concentatons aound a ccula hole o ccula ncluson n a be-enoced vscoelastc plate unde unom shea. Allam and Zenkou [1] have used the small paamete method as well as the method o eectve modul o the bendng esponse o a beenoced vscoelastc ached bdge model wth quadatc thckness vaaton and subjected to unom loadng. In [13] they have also obtaned the stesses aound lled and unlled ccula holes n a be-enoced vscoelastc plate unde bendng. The same autho [14] have developed closed om solutons o the otatng beenoced vscoelastc sold and annula dsks wth vaable thckness by applyng the genealzaton o Illyushn's appoxmaton method. In addton Allam et al. [15] have detemned the stess concentatons aound a tangula hole n a be-enoced vscoelastc composte plate unde unom tenson o pue bendng. Also Zenkou et al. [16] have pesented the elastc and vscoelastc solutons to otatng unctonally gaded hollow and sold cylndes. In the pesent pape the otatng be-enoced vscoelastc hollow cylnde s analytcally studed. The thckness o the cylnde and the elastc popetes ae Copyght 010 ScRes.

2 490 A. M. ZENKOUR taken to be unctons n the adal coodnate. The govenng second-ode deental equaton s deved and solved wth the ad o some hypegeometc unctons. The dsplacement and stesses o otatng beenoced vscoelastc nhomogeneous othotopc hollow cylnde wth vaable thckness and densty subjected to vaous bounday condtons ae obtaned. Specal cases o the studed poblem ae establshed and numecal esults ae pesented n gaphcal oms.. Rotaton o Vscoelastc Cylndes Accodng to the elastc soluton gven n pat I we can use the method o eectve modul and Illyushn's appoxmaton method to solve the otaton poblem o vaable thckness and densty vscoelastc hollow cylnde enoced wth undectonally elastc bes. Fo an othotopc cylnde the complance paametes j can be expessed n tems o the engneeng chaactestcs as [17]: n whch 1 E 1 E z z z z 11 = = E zz E z z 1 = = E z z Ez z z 13 = = E z z Ez z z 3 = = (1) =1 () z z z z z z whee E ae Young's modul and j ae Posson's atons whch ae elated by the ecpocal elatons: z z z z = = =. (3) E E E Ez E Ez Now consde a hollow cylnde made o a composte mateal composed o two components. A vscoelastc mateal as a st component enoced by undectonal elastc bes as a second component. The st o these components plays the ole o lle and may posses the popetes o a lnea vscoelastc mateal and t s descbed by the modulus E and Posson's ato. The othe component wll be seve as the enocement and s an elastc mateal wth modulus o elastcty E and Posson's ato. Unde the above consdeatons and usng the method o eectve modul [1418] Young's modul and Posson's atos wth = = z = z = 1 and = = ae gven by [19]: z z EE E = E = Ez = E1 E E 1E 1 EE = = 1 E E E E (4) whee s the volume acton o be enocement. Thus t s obvous that the ecpocal elatons gven n Equaton (3) ae ullled. Note that the vscoelastc modulus E s gven by: 9K E = (5) whee K s the coecent o volume compesson (the bulk modulus) and t s assumed to be not elaxed.e. K = const. and s the dmensonless kenel o elaxaton uncton whch s elated to the coespondng Posson's ato by the omula: 1 =. (6) 1 Substtutng om Equatons (5) and (6) nto Equaton (4) yelds 9 p 91 p E = E = E Ez = E 1 1 9p 119 p = = 91 p 9 p 1 (7) o n the smple om 9Ep E = E = 1 g E z = E 9p 1 1 g p 1 1 g 1 g = 9p11g 9p 1g = 11 g 1 (8) n whch p g = 1 = = 1 (9) 1 1 whee p = K / E s the consttutve paamete. Wth the help o Equatons (1) and (8) one can ewte the solutons gven n Pat I o ths pape; see Equatons (0) and (3)-(5); n the om: u = u = ˆ (10) ˆ = = ˆ zz zz whee Copyght 010 ScRes.

3 A. M. ZENKOUR b ˆ = = 0 b. (11) E It s to be noted that n elastc compostes the adal dsplacement and stesses ae unctons o and whle n vscoelastc compostes they ae opeato unctons o the tme t and. Accodng to Illyushn's appoxmaton method [11190] the uncton u can be epesented n the om: 5 u ( )= A (1) =1 whee ( ) ae some known kenels constucted on the base o the kenel and may be chosen n the om: 1 1 =1 = 3 = = 4 = g 1 5 = g (13) whee g ( =1) ae gven n Equaton (9). The coecents A ( ) ae detemned om the system o algebac equatons whee 5 j=1 LA j j = B =1 5 (14) 1 1 (15) L = d B = u d. j 0 j 0 Now let us consde the elaxaton uncton n an exponental om t = c1 ce t (16) whee c 1 and c ae constants to be expementally detemned. Laplace-Cason tansom can be used to detemne the unctons () t and g () t. Denotng the tansoms o () t and g () t by () t and g () t snce the tansom o () t s sc ()= s c1 (17) s thus we get 1 c c1 / c1c t = 1 e = t c1 c1 c 1 c 1 c1/ 1c 1c g t = 1 e 1c1 1 c1c (18) Equaton (1) o a vscoelastc composte may be ecoded to obtan explct omula o the adal dsplacement as uncton o and tme t n the om: t u ( t)= A( ) ( t) A ( ) ( t) d ( ) 1 0 t t t 5( ) ( ) ( ). 0 A ( ) ( t) d ( ) A ( ) g ( t) d ( ) A g t d (19) Takng ()= t 0H() t whee H () t s the Heavsde's unt step uncton gven by 1 t 0 Ht ()= 0 t < 0. Then Equaton (19) takes the om (0) u ( t)= 0A1Ht () A() t A3() t Ag 4 () t Ag 5 () t. 1 (1) whee () t () t and g () t ae gven n Equatons (16) and (18). Usng the same technque once agan to obtan the adal ccumeental and axal stesses o the otatng be-enoced vscoelastc hollow cylnde wth vaable thckness and densty by eplacng only u ( t) wth ( t) and makng the sutable changes n ths case. 3. Applcatons In ths secton some numecal examples o the otatng be-enoced vscoelastc nhomogeneous vaablethckness cylnde wll be ntoduced. The esults o the pesent poblem wll be gven o thee sets o geometc paametes k and n o the thckness pole. The numecal applcatons wll be caed out o the adal dsplacement and stesses that beng epoted heen ae n the ollowng dmensonless oms: u = = = =. u zz ˆ ˆ z ˆ The eect o the elastc popetes o the cylnde consttutve and tme paametes on the dmensonless adal dsplacement and stesses wll be shown. The calculatons wll be caed out o the ollowng values o paametes: =0.3 = c1 =0.1 c =0.9 and =0.5. In addton othe paametes ae taken (except othewse stated) as: p = 0. k =.5 n= 0.8 and m =1. Also the coecent s stll unknown and the tme paamete ( t) s gven n tems o t. The dstbutons o the dmensonless stesses and dsplacement though the adal decton o the otatng be-enoced vscoelastc nhomogeneous vaablethckness cylnde ae plotted n Fgues 1-3 accodng to the FF CC FC and CF bounday condtons espectvely. Fo all hollow cylndes the dmensonless adal dsplacement u s the lagest n the same poston o small k.e. k =0.6. Fo FF and CF hollow cylndes the dmensonless stesses ae the lagest o small n. The mnmum values o the dmensonless adal stess at the oute suace o the CC and FC hollow cylndes ae lage o k =0.6. Also Copyght 010 ScRes.

4 49 A. M. ZENKOUR Fgue 1. Dmensonless stesses and dsplacement o a vaable-thckness vscoelastc hollow cylnde subjected to vaous bounday condtons (k = 0.6 n = 0.8 m = 0.5). Copyght 010 ScRes.

5 A. M. ZENKOUR 493 Fgue. Dmensonless stesses and dsplacement o a vaable-thckness vscoelastc hollow cylnde subjected to vaous bounday condtons (k =.5 n = 0.8 m = 0.5). Fgue 3. Dmensonless stesses and dsplacement o a vaable-thckness vscoelastc hollow cylnde subjected to vaous bounday condtons (k =.5 n = 0.4 m = 0.5). Copyght 010 ScRes.

6 494 A. M. ZENKOUR the dmensonless ccumeental and axal z stesses ae smalle though the adal decton o the CC hollow cylndes when k =0.6. The maxmum value o at the nne suace o FC hollow cylnde ae the smallest when k =.5 and n = 0.8. In addton the d- mensonless axal stesses ae monotone deceasng n and t s smalle o n = 0.4 than o n = 0.8. Fo a pole wth geometc paametes k =.5 and n =0.8 the dmensonless dsplacement and stesses ae plotted n Fgues 4-7 o the otatng be-enoced Fgue 4. Dstbuton o dmensonless stesses and dsplacement though the adal decton o a FF vaablethckness vscoelastc hollow cylnde. Fgue 5. Dstbuton o dmensonless stesses and dsplacement though the adal decton o a CC vaablethckness vscoelastc hollow cylnde. Copyght 010 ScRes.

7 A. M. ZENKOUR 495 Fgue 6. Dstbuton o dmensonless stesses and dsplacement though the adal decton o a FC vaablethckness vscoelastc hollow cylnde. vscoelastc nhomogeneous cylnde subjected to vaous bounday condtons wth deent values o the paamete m. The stesses and dsplacement o m =1 ae the smallest when compaed to the esults o m =0 Fgue 7. Dstbuton o dmensonless stesses and dsplacement though the adal decton o a CF vaablethckness vscoelastc hollow cylnde. and 1. Fo FF and FC hollow cylndes the dmensonless adal dsplacement u has changed concavty. The dmensonless adal stess nceases stly to get ts maxmum value then t deceases agan at the Copyght 010 ScRes.

8 496 A. M. ZENKOUR extenal suace to get zeo value o FF bounday condton whle t tends to a constant value o FC bounday condton. In both cases the dmensonless ccumeental stess has maxmum value at the nne suace. Also the dmensonless adal dsplacement u nceases dectly as the dmensonless adus nceases o CF hollow cylndes whle the hghest values o t occu nea the extenal suaces o the CC hollow cylndes. The dmensonless adal stess s monotone deceasng n o CC and CF hollow cylndes. In all gues the dmensonless axal stess z deceases om the nne to the oute suace. Also the dmensonless adal dsplacement o a pole k =0.6 n =0.8 and m =1s plotted n Fgue 8 wth vaous values o the consttutve paamete p. Fo CF and FF hollow cylndes the dmensonless adal dsplacement u and the concavty changed o t o FC hollow cylnde ncease wth the deceasng o the consttutve paamete p. In addton the maxmum values o u decease wth the ncease o p o CC hollow cylnde. Note that the maxmum values o u occu at the same poston =0.7 o deent values o p. Fnally the nluence o tme paamete on the dmensonless dsplacement and stesses o vaable thckness vscoelastc hollow cylnde subjected to FF CC FC and CF bounday condtons s plotted n Fgue 9. Ths nluence s studed at the poston =0.5 wth geometc paametes k =0.6 n =0.8 and m =1. Fo all hollow cylndes the dmensonless adal dsplacement u nceases apdly wth nceasng the tme paamete to get a constant value o 55. Also o FF hollow cylndes the dmensonless adal and ccumeental stesses may be unchanged wth tme paamete.5 whle the dmensonless axal stess z nceases apdly to stll unchanged o 8. Fo CC and FC hollow cylndes the hghest values o and z occu at 3.5 and 5 espectvely then they ae deceasng n the ntevals 3 < < 14.5 < < 16 and 5< <17 to stll unchanged o and 17 espectvely. Also o CF hollow cylnde the mnmum value o the dmensonless adal stess happens at then t s nceasng slowly to app- Fgue 8. The eect o the consttutve paamete p on u o a vaable-thckness vscoelastc hollow cylnde. Copyght 010 ScRes.

9 A. M. ZENKOUR 497 Fgue 9. The eect o tme paamete on (a) cylnde at = 0.5. u (b) (c) and (d) z o a vaable-thckness vscoelastc hollow oach a constant value o 13. Howeve the dmensonal ccumeental and axal stesses ncease to get the maxmums at 3.5 and 7.5 espectvely then decease to stll unchanged o 15 and 17.5 espectvely. 4. Conclusons The otaton poblem o a be-enoced vscoelastc nhomogeneous vaable-thckness hollow cylnde has been studed. The elastc poblem s solved analytcally by usng the hypegeometc unctons. The vscoelastc poblem s solved usng both the method o eectve modul and Illyushn's appoxmaton method. Analytcal soluton o otatng be-enoced vscoelastc nhomogeneous ansotopc hollow cylnde o vaable thckness and densty subjected to deent bounday condtons ae deved. The dsplacement and stesses o otatng be-enoced vscoelastc homogeneous sotopc hollow cylnde wth unom thckness and densty ae obtaned as specal cases o the nvestgated poblem. The eects due to many paametes on the adal ds- placement and stesses ae nvestgated. 5. Reeences [1] I. E. Toyanovsk and M. A. Koltunov Tempeatue Stesses n a Long Hollow Vscoelastc Cylnde wth Vaable Inne Bounday Mekhanka Polmeov Vol. 5 No pp [] M. A. Koltunov and I. E. Toyanovsk State o Stess o a Hollow Vscoelastc Cylnde Whose Mateal Popetes Depend on Tempeatue Mechancs o Composte Mateals Vol. 6 No pp [3] E. C. Tng and J. L. Tuan Eect o Cyclc Intenal Pessue on the Tempeatue Dstbuton n a Vscoelastc Cylnde Intenatonal Jounal o Mechancal Scences Vol. 15 No pp [4] L. Kh. Talybly Deomaton o a Vscoelastc Cylnde Fastened to a Housng unde Non-Isothemal Dynamc Loadng Jounal o Appled Mathematcs and Mechancs Vol. 54 No pp [5] W. W. Feng T. Hung and G. Chang Extenson and Toson o Hypevscoelastc Cylndes Intenatonal Jounal o Non-Lnea Mechancs Vol. 7 No pp Copyght 010 ScRes.

10 498 A. M. ZENKOUR [6] V. G. Kanaukhov and I. K. Senchenkov Themome- Chancal Behavo o a Vscoelastc Fnte Ccula Cylnde unde Hamonc Deomatons Jounal o Engneeng Mathematcs Vol. 46 No pp [7] D. Bland The Lnea Theoy o Vscoelastcty Pegamon Pess New Yok [8] D. Abolnsh Elastcty Tenso o Undectonally Renoced Elastc Mateal Polyme Mechancs Vol. 4 No pp [9] A. A. Illyushn and B. E. Pobeda Foundatons o Mathematcal Theoy o Themo-Vscoelastcty n Russan Nauka Moscow [10] M. N. M. Allam and P. G. Appleby On the Plane Deomaton o Fbe-Renoced Vscoelastc Plates Appled Mathematcal Modellng Vol. 9 No pp [11] M. N. M. Allam and P. G. Appleby On the Stess Concentatons aound a Ccula Hole n a Fbe-Renoced Vscoelastc Plate Res Mechanca Vol. 19 No pp [1] M. N. M. Allam and A. M. Zenkou Bendng Response o a Fbe-Renoced Vscoelastc Ached Bdge Model Appled Mathematcal Modellng Vol. 7 No pp [13] A. M. Zenkou and M. N. M. Allam Stesses aound Flled and Unlled Ccula Holes n a Fbe-Renoced Vscoelastc Plate unde Bendng Mechancs o Advanced Mateals and Stuctues Vol. 1 No pp [14] A. M. Zenkou and M. N. M. Allam On the Rotatng Fbe-Renoced Vscoelastc Composte Sold and Annula Dsks o Vaable Thckness Intenatonal Jounal o Computatonal Methods n Engneeng Scence and Mechancs Vol. 7 No pp [15] M. N. M. Allam A. M. Zenkou and H. F. El-Mekawy Stess Concentatons n a Vscoelastc Composte Plate Weakened by a Tangula Hole Composte Stuctues Vol. 79 No pp [16] A. M. Zenkou K. A. Elsba and D. S. Mashat Elastc and Vscoelastc Solutons to Rotatng Functonally Gaded Hollow and Sold Cylndes Appled Mathematcs and Mechancs - Englsh Edton Vol. 9 No pp [17] A. E. Bogdanovch and C. M Pastoe Mechancs o Textle and Lamnated Compostes wth Applcatons to Stuctual Analyss Chapman and Hall New Yok [18] B. E. Pobeda Stuctual Ansotopy n Vscoelastcty Mechancs o Composte Mateals Vol. 1 No pp [19] A. M. Zenkou Themal Eects on the Bendng Response o Fbe-Renoced Vscoelastc Composte Plates usng a Snusodal Shea Deomaton Theoy Acta Mechanca Vol. 171 No pp [0] M. N. M. Allam and B. E. Pobeda On the Soluton o Quas-Statcal Poblems o Ansotopc Vscoelastcty Izv. Acad. Nauk ASSR In Russan Mechancs Vol. 31 No pp Copyght 010 ScRes.