Analysis of five-parameter Viscoelastic model under Dynamic Loading

Transcription

1 J. cad. Idus. Res. Vol. (7) December 49 RESERCH MNUSCRIPT ISSN: 78-5 alysis of five-parameter Viscoelastic model uder Dyamic Loadig Rajeesh Kakar K. Kaur ad K.C. upta Pricipal DIPS Polytechic College Hoshiarpur-46 Idia Faculty of Sciece MSCE Muktsar-56 Idia bstract The purpose of this study is to aalysis the viscoelastic models uder dyamic loadig. five-parameter model is chose for study exhibits elastic viscous ad retarded elastic respose to shearig stress. The viscoelastic specime is chose which closely approximates the actual behaviour of a polymer. The module of elasticity ad viscosity coefficiets are assumed to be space depedet i.e. fuctios of x i o-homogeeous case ad stress-strai are harmoic fuctios of time t. The expressio for relaxatio time for five parameter viscoelastic model is obtaied by usig costitutive equatio. The dispersio equatio is obtaied by usig Ray techiques. The model is justified with the help of cyclic loadig for maxima or miima. Keywords: Shear waves viscoelastic media shearig stress asymptotic method dyamic loadig. Itroductio The vibratios i earthquakes are due to differeces i dyamic characteristics therefore the cyclic stress-strai behaviour of material play a vital role for reliable predictio of the seismic respose. May researchers studied structural poudig durig earthquakes. The lack of iformatio cocers multi-dimesioal waves i viscoelastic-media ad i particular for o-homogeous media therefore a formal study of o-homogeeous viscoelastic models uder dyamic loadig is preseted. Jakowski (5) discussed the liear viscoelastic model ad the oliear viscoelastic model. agostopoulos (988) studied the liear viscoelastic model of collisio to simulate structural poudig. Jakowski et al. (998) studied the poudig of superstructure segmets i bridges with the help of liear viscoelastic model. Muthukumar ad DesRoches (6) made a compariso study usig two sigle degree of freedom (SDOF) systems for capturig poudig. Westermo (989) suggested likig buildigs with beams which ca trasmit the forces betwee them elimiatig dyamic cotacts. I this study the module of elasticity ad viscosity coefficiets are assumed to be space depedet i.e. fuctios of x. Further shearig strai ad stress are i t take as harmoic fuctios of time t i.e. e ad i t it e e. The expressio for relaxatio time for five parameter viscoelastic model is obtaied by usig costitutive equatio. The dispersio equatio is obtaied by usig Ray techiques. The model is justified with the help of cyclic loadig for maxima or miima. Methodology The assumptios chose are such that the coclusios draw o the basis of which agree quite reasoably ad closely with the observed results of experimetal tests. Followig are the pricipal assumptios ad hypothesis o which the problem has bee costructed.. Homogeeity: The material of a structure to be cosidered should be homogeeous i structure ad cotiuous at all poits of the body. homogeeous structure meas that ay how so ever small particle/portio of the body uder cosideratio must possess the same properties. mog the materials that are cosidered to be homogeeous are metals alloys such as steel alumium copper etc.. bsolutely elastic: The bodies cosidered beig absolutely elastic with respect to deformatio whe their deformatios which appear due to exteral force completely disappear upo removal of the load. ctually this holds true up to a defiite value of load.. Isotropy: Material cosidered is take to be isotropic whe it possesses the same characteristic i all directios. Isotropic materials iclude metals cocrete ad some plastics. Materials possessig differet properties i various directios are called a isotropic. Examples are wood reiforced plastic etc. 4. Ifiite small deformatios: Whe deformatios of elastic bodies uder the actio of exteral loads are small as compared with the dimesios of the bodies i.e. the dimesios/shape are ot chaged substatially o elastic deformatios. This assumptio simplifies substatially the calculatio sice it makes possible to eglect chages i the arragemet of the forces o deformatio. Youth Educatio ad Research Trust (YERT) Rajeesh Kakar et al.

2 J. cad. Idus. Res. Vol. (7) December 4 5. Super-positio-priciple: Sice the deformatio cosidered beig small it ca be assumed that exteral forces act idepedetly from oe-aother i.e. the deformatios ad iteral forces appearig ielastic bodies do ot deped o the order i which the exteral forces are applied. esides it is assumed that the total effect of the whole system of forces actig o the body is the sum of the effects produced by idividual forces. bout the model It is a five parameter model with two sprigs S ( )S ( ) with module of elasticity ad three dash pots D (η ) D (η ) D (η ) with viscosities η η η. It has three sectios. Sectio I Cotais oe sprig S ( ) Sectio II cotais three elemets oe sprig S ( ) ad two dash pots D (η ) D (η ) where the sprig S ( ) ad dash-pot D (η ) are i series formig Maxwell l-model ad the dash pot D (η ) is parallel to the Maxwell elemet. The sectio III cotais oly oe dash-pot D (η ). The sprig represets recoverable elastic respose ad dash pots represet elemets i structure givig rise to the viscous drag/dissipative respose (where the viscosity of the oil/fluid i the dash pot decreases with the icrease i temperature). Sectio-I: Represeted by oly oe sprig S ( ) represets the elastic regio (glassy) which is domiat at low temperatures. I this rage of behavior of the material a applied stress (load) produces a strai which is reversible upo the release of the stress uder the elastic limits (istataeous deformatio). I case of polymer materials the strai is due to the stretchig of bods withi ad betwee molecular chais. The chais which are froze to-gather iitially caot flow past each other ad may oly be separated by fracture which i our case does ot happe as we are cosiderig small-deformatios. Thus the sprig S ( ) represets the behavior of polymer i glass regio. Sectio-II: It represets leathery ad rubbery regio. I the leathery regio the modulus of elasticity drops rapidly with load (temperature) ad reversible slidig becomes possible i short segmets of the chais of macromolecules. Small sectios move ad the cause the eighborig sectios to move co-operatively. Here a trasitio appears betwee the elastic behaviors of Sectio I ad viscoelastic behavior of Sectio II. The reversibility of the movemets of the short chai segmets is expressed by the sprig S ( ) i Sectio II ad the resistace to this movemet by the dash-pots D (η ) ad D (η ). I the rubbery phase the Viscoelastic behavior i sectio-ii domiates the deformatio. s the load icreases the molecular segmets slide reversibly past oe aother ad ted to straighte out i the directio of the load. Sectio-III: This sectio is represeted by a sigle dashpot D(η ) where permaet molecular slidig domiates the deformatio process. t the higher loadig the viscosity decreases due to iteral fractios which give rise to temperature icrease ad apparet modulus also drops eve to such a extet the material behaves as fluid as i the case of glaciers or melts gels etc. Middle sectio-ii Which is a series combiatio with a sprig S ( ) of sectio-i ad a dash-pot D (η ) sectio-iii ca be geerated from Voigt Model by addig oe more dash-pot to the sprig side so that it becomes a Maxwell-Model or it ca be degeerated from a parallel combiatio of two Maxwell elemets by detachig oe sprig from oe of the Maxwell elemet i.e. takig the modulus of elasticity i this Maxwell elemet as ifiitely greater i.e.. Sice i sectio-ii the dash-pot D (η ) is fill to flow as is ot restricted by ay sprig so the model exhibits log term viscous flow. The Viscous elemet D (η ) ad the Maxwell elemet (D (η ) S ( ) ) possess this property. Durig relaxatio the dash-pot D (η ) which is free from the restrictios of a sprig will evetually take up the whole extesio ad stress will drop to zero slowly ad ultimately. It is further added that the combiatio/etwork of elastic elemets (Sprig S()) ad viscous elemet (dashpots D(η)) Maxwell-model Voigt-Model is uidirectioal i.e. all the elemets lie i the same directio ad all cocered forces ad deformatios act i this directio ad are i the same plae. Costitutive relatio for five parameter model The five parameter model cosists of two sprigs S ( ) S ( ) Where s ad s are the module of elasticity associated to them ad three dash-pots D ( ) D ( ) D ( ) where are the Newtoia Viscosity coefficiets associates to these elemets. The module of elasticity ad viscosity coefficiets are assumed to be space-depedet i.e. fuctios of x i ihomogeeous case take ito cosideratio. Uidirectioal problem is formed by takig the material i the form of filamet of o-homogeeous viscoelastic material by takig oe ed at x. The co-ordiate x is measured positive i the directio of the axis of the filamet. Time is specified by t ad u respectively specify the oly o-zero compoets of stress shearig strai ad particle displacemet. The modal has bee divided ito three sectios. Sectio I cotais oe sprig S ( ) ad sectio II cotais two dash-pots D ( ) D ( ) ad oe sprig S ( ). Youth Educatio ad Research Trust (YERT) Rajeesh Kakar et al.

3 J. cad. Idus. Res. Vol. (7) December 4 Fig.. Five parameter viscoelastic model. The stress-strai relatio for the model represetig the Viscoelastic body for total stress ( ) ad strai ca be obtaied from Eq. () Eq. (4) Eq. (5) ad Eq. (6) as: D D D D (6) Now we take ij ij i j Si( i ) Dj( ) j (7) Where S ( ) elastic modulus of sprig ad i i Sectio III cotais oe dash-pot D (η ). The etwork of coectio of various dash-pots ad sprigs is represeted i figure. Uder the supper-suppositio priciple strais are added i the case of series coectios ad stresses are added whe they are i parallel. Now if be the three shearig strais elogatios i respective sectios coected i series the total elogatio is give by + + () The total Stress i the etwork remais the same. I each sectio but i the case of sectio II which is sub-divided ito two sectios is added i.e. σ σ + σ where σ ad σ are the stresses i the sub-sectios as show i figure. Relatio for stress ad strai for D ) for sectio II is represeted by sigle dash-pot. σ () Sice the sub-sectio II is represeted by a Maxwell- elemet the the relatio is expressed as (D ( ) + η ) D ( ) () Sice σ σ + σ for Sectio II therefore (D ( ) + η ) ( η D D D For sectio I for the Sprig S ( ) the stress-strai relatio is give by (5) For sectio III; for the dash-pot D ( ) the stress strai relatio is give by (6) (4) Dj( j ) viscosity of dash-pot ij j ( i ; j ) i Usig Eq. (7) ad Eq. (8) we get D D D D. (8) Put R R R. Eq. (9) we get i D R R D R R D R D D R D () The Eq. () ca be writte i terms of differetial operator form as m m m b x t D x t () Where the order m ad of sums o R.H.S ad L.H.S i the relatio () depeds upo the structure of the mechaical model represetig the viscoelastic body. ad m are the combiatios of the material costats ad R R + R R R d D. dt Eq. () is the required differetial operator form of costitutive relatio for the model for viscoelastic material to be studied. overig equatios for viscoelastic model Oe of the goverig equatio for the viscoelastic model is costitutive relatio ad is () f t tt t tt t tt t (9) (a) Youth Educatio ad Research Trust (YERT) Rajeesh Kakar et al.

4 J. cad. Idus. Res. Vol. (7) December 4 The equatio of motio is x u tt or x log x x The displacemet-strai relatio is u x The shearig stress field is + t ttt tt log x x x tt log t x xt () (4) + (5) Solutio for five parameter viscoelastic model We assume that the solutio σ (x t) of Eq. (5) may be represeted by the series x t F t h x F x x t h x F F F h F F F With x x t F F F F t tt ttt (6) The various derivatives stress with respect to x ad t are F h F x x F h h F + h x F x t F h x h F h x F xt F h x F (7) From Eq. (6) ad Eq. (7) F F F h ( log ) x x h x h x h ( log ) h x x log x F F F h x log h x Comparig the Coefficiet of F we get log x log x h F x (8) (9) Comparig the Coefficiet of F we get h ( log ) h h log x x x x F Comparig the coefficiet of we get h x ( log) x h x h x h Comparig the Coefficiet of F we get h Let ad h x h the Eq. () reduces to () () () () From Eq. (9) ad Eq. () we get h x( log) x h x h (4) From Eq. () ad Eq. (4) h x h x. h x. From Eq. () we get (5) S (6) Where S h x( log) x h x h From Eq. (a) we get h x S From Eq. (6) ad Eq. (7) h S x ad S. (7) Youth Educatio ad Research Trust (YERT) Rajeesh Kakar et al.

5 J. cad. Idus. Res. Vol. (7) December 4 R Takig R R -. R ( R + R ) R R + R ad or ( R R ) ad fially we get R R R R (8) Eq. (8) is the expressio for relaxatio time for five parameter viscoelastic model. Dyamic loadig The time parameter t is itroduced ito a experimetal scheme i dyamic experimets by cyclic deformatio of the specime. The frequecy of the oscillatios plays the role of the time factor.the cyclic deformatio is the fudametal process of determiig the metioed characteristics. The greatest preferece is give to harmoic oscillatios. Harmoic actio of the stress/strai produces a correspodig harmoic respose i the strai/stress. Let us cosider that shearig strai ( ) iduced i elastic body which ca be expressed by a harmoic actio as: e (9) Where is the amplitude is the frequecy of oscillatios ad t is the time. ccordig to Hooke s Law stress is e e Where at t () For a elastic body the strai ad stress vary harmoically ad there is o lack i the harmoic motio i phase as both have e as a factor. Thus a elastic body respods istataeously to the exteral actio (strai/stress). The phase shift agle betwee strai ad stress is zero. For a ideal viscous body the Newto s Law of flow to a fluid body is as: D ie Where is the Viscosity of the body ad at t () Thus for a viscous deformatio stress advaces by the strai by a phase agle. Thus the phase shift agle for the stress-strai uder periodic harmoic deformatio for elastic body is also for the viscous body it is. Therefore the phase shift agle for the viscoelastic body must be betwee zero ad i.e.. The laggig i phase of the strai behid the stress is due to the presece of relaxatio processes i the case of viscoelastic body as phase shift agle is give by. Hece e i( wt ) e () If we represet the projectio of the stress vector o axis of co-ordiates by takig x ad y we write i () Where ad represet that real ad imagiary parts respectively. If the strai is iitially set harmoically the the strai vector e coicides with its real part ad imagiary part The the modulus of Viscoelastic body with harmoic loadig ca be writte as The phase agle is give as ta i i (4) (5) I the case of preset model (Five-Parameter; two sprigs (S ( ) S ( )); three dash-pots D ( ) D ( ) D ( ) which represets a liear viscoelastic behavior uder a give actio of loadig the stress is directly proportioal to strai. This is also true for time depedet stress ad strai relatio i.e. for viscoelastic body the stress is t ( i ) t Where e e (6) Youth Educatio ad Research Trust (YERT) Rajeesh Kakar et al.

6 J. cad. Idus. Res. Vol. (7) December 44 Usig the relatio for a elastic body the costitutive relatio for the physical state represetig the five parameter model is give by D ( ) D ( ) D D (7) 4 Where 4 Let i (8) Usig Eq. (6) Eq. (7) ad Eq. (8) we get i ( ) 4 i O solvig we get ( i ) i i ( ) e e (9) 4 4 i i i i i i 4 4 Separate Eq. (4) ito real ad imagiary parts we get 4 4 d Loss taget is give by 4 Ta (4) (4) (4) (4) i To fid the values of we put c D Where 4 c D Now D ( ( i )) (44) C D 4 (45) With the help of Eq. (45) the dispersio relatio ca be derived (calculatios are show i the appedix) 6 4 C D ( C) D C C Where C D 4 4 c D (46) (47) Equatio (46) gives the dispersio equatio for wave propagatio. It is a cubic i givig three roots it must have oe real root as complex roots always occur i cojugate pairs or all three roots are real for has either a maximal value or miimum value. Therefore takig roots as Sum of roots D we get ( C) (48) Product of roots take two at a time D C C Products of roots C To determie (49) (5) through relatio (47) seems ot to be so easy but if we observe carefully the value of we ca coclude about the roots as follows (see appedix): 4 4 ( ) (5) Youth Educatio ad Research Trust (YERT) Rajeesh Kakar et al.

7 J. cad. Idus. Res. Vol. (7) December 45 To fid the other two roots for the Eq. (46) from Eq. (48) Eq. (49) ad Eq. (5) such that (Takig oe of the values for as ) C for the extreme values of D C. C The Eq. (46) ca be expressed as C x D ( C) x We get the roots C D C C D C C (5) D C (5) Where 4. C 4. D (54) Error due to approximatio is 4 C D C Usig Eq. (49) Eq. (5) ad Eq. (48) oe ca fid C C D C From Eq. (49) D C /. D C C D C C (55) (56) pproximate value C C C D C D C (57) Error ca be calculated by subtractig Eq. (56) ad Eq. (57) Error (ξ) D - C C C C D C D C (ξ) C D ( C) C D C C D C Takig the +ve sig we get C D C D ( C) D C C D C Case- t very low frequecies (from Eq. (5)) 4 ( ) The it is to be iferred that durig the cyclic loadig iitially (58) i.e. there must be a poit of maxima or miima betwee ad 4 Case- t very high frequecies (from Eq. (5)) ( ) (59) The it is to be iferred that durig the cyclic loadig iitially i.e. there must be a poit of ad 4 maxima or miima betwee but for 4 it is observed that for C there must be a poit of maxima as whe iitially () icreases from zero to maximum value ( 4) m ad agai states that dimiishig ad reaches zero at 4 which justifies for the model for the Viscoelastic materials. Youth Educatio ad Research Trust (YERT) Rajeesh Kakar et al.

8 J. cad. Idus. Res. Vol. (7) December 46 Loadig of the model Whe relaxatio is applied to the model i.e. the model is uder the ifluece the costat deformatio the specime represetig the model is deformed to the give strai ad after which it is maitaied costat where as the stress required to maitaied these strais ( ) dimiishes at e costat uder thermal coditios. The Costitutive equatio uder costat deformatio (strai) e costat reduces to D D (6) Where ( ) ( ). Eq. (6) ca be solved we takig the roots of the auxiliary equatio as m ad m. (6) Where ad are relaxatio times of the specime. mm. ( ) From Eq. (6) the Eq. (6) becomes D m m D mm ( ) (6) (6) The Solutio of Eq. (6) is m t m t t e e (64) To elimiate ad t t Eq. (64) reduces to Hece d dt (65) Coclusio From the above study the followig coclusios ca be doe: Durig the cyclic loadig iitially there must be a poit of maxima or miima betwee ad 4.. For sufficietly large relaxatio time the stress i the specime will drop to zero.. The phase shift agle for the viscoelastic body must be betwee zero ad. ckowledgemets The authors are thakful to the referees for their valuable commets. Refereces. agostopoulos S Poudig of buildigs i series durig earthquakes. Earthquake Egg. Structural Dyamics. 6(): Jakowski R. Wilde K. ad Fujio Y Poudig of superstructure segmets i isolated elevated bridge durig earthquakes. Earthquake Egg. Structural Dyamics. 7: Muthukumar S. d Desroches R. 6. Hertz cotact model with oliear dampig for poudig simulatio. Earthquake Egg. Structural Dyamics. 5: Westermo.D The dyamics of iterstructural coectio to prevet poudig. Earthquake Egg. Structural Dyamics. 8: Jakowski R. 5. No-liear viscoelastic modellig of earthquake-iduced structural poudig. Earthquake Egg. Structural Dyamics. 4(6): ppedix m m (66) Where m m m m m m m m m m mt mt m t m t t me me me me t so that For sufficietly large time. Hece with loger periods of observatio the stress i the specime will drop to zero i.e. equilibrium state will be achieved. Youth Educatio ad Research Trust (YERT) Rajeesh Kakar et al.