RHEOLOGICAL RESPONSES OF VISCOELASTIC MODELS UNDER DYNAMICAL LOADING

Transcription

1 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN RHEOLOICL RESPONSES OF VISCOELSTIC MODELS UNDER DYNMICL LODIN Rajeesh Kakar, Ph.D., Correspodig author Pricipal, DIPS Polytechic College, Idia Kawaljeet Kaur Kisha Chad upta ssistat Professor, BMSCE, Idia bstract The purpose of this paper is to study the rheological resposes of four ad fiveparameter viscoelastic models uder dyamic loadig. These models are chose for studyig elastic, viscous, ad retarded elastic resposes to shearig stress. The viscoelastic specime is chose which closely approximates the actual behavior 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 complex viscosity of five parameter model is calculated. The expressio for relaxatio time for five parameters ad four parameter viscoelastic models have bee obtaied by usig a costitutive equatio. The dispersio equatio is obtaied by usig Ray techiques. The rheological resposes for both models uder dyamical loadig are show graphically. lso, the five parameter model is justified with the help of cyclic loadig for maxima or miima. Keywords: Shear Waves, Viscoelastic Media, symptotic Method, Dyamic Loadig, Friedlader Series Itroductio The viscoelasticity theory is used i the field of solid mechaics, seismology, exploratio geophysics, acoustics ad egieerig. The solutios of may problems related to wave-propagatio i homogeeous media are available i may literatures of cotiuum mechaics of solids. However, i the recet years, the iterest has arise to solve the problems coected with o-homogeeous bodies. These problems are useful to uderstad the properties of polymeric materials ad idustrial related applicatios. The practice ad 87

2 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN theoretic aalysis idicate that the dyamic loadigs (such as earthquake, tsuami, ragig billow ad vibratio) are the importat factor ad maily the power of iducig geological disaster of soft rock-soil. The rheological mechaics respose ad rheological parameters of specime are tested ad aalyzed uder dyamic loadig, the viscoelastic-plasticity rheological dyamic model is established, ad ew rheological equatio is deduced. The vibratios i earthquakes are due to differeces i dyamic characteristics therefore the cyclic stress-strai behavior 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-homogeeous media, therefore, a formal study of o-homogeeous viscoelastic models uder dyamic loadig is preseted. Modelig ad model parameter estimatio is of great importace for a correct predictio of the foudatio behavior. May researchers lfrey (944), Barbera ad Herrera (966), chebach ad Reddy (967), Bhattacharya ad Segupta (978), ad charya et al. (8) formulated ad developed this theory. Further, Bert ad Egle (969), bd-lla ad hmed (996), Batra (998) successfully applied this theory to wave-propagatio i homogeeous, elastic media. Murayama ad Shibata (96), Schiffma et al. (964) have proposed higher order viscoelastic models of five ad seve parameters to represet the soil behavior. Jakowski et al. (998) 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. (5) studied the poudig of superstructure segmets i bridges with the help of a liear viscoelastic model. Muthukumar ad DesRoches (6) did 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. Kakar et al.; () ad Kaur et al.; () aalyzed viscoelastic models uder Dyamic Loadig, Recetly, Kakar ad Kaur (3) aalyzed five parameter model of the propagatio of cylidrical shear waves i o-homogeeous viscoelastic media. The behavior of real materials caot be completely represeted by the simple Maxwell ad Kelvi Model. More complicated models are required with a greater flexibility i portrayig the respose of actual material. Maxwell uit i parallel with a sprig is the stadard liear solid ad a Maxwell uit i parallel with a dashpot is the viscous model. four parameter model cosistig of two sprigs ad two despots may be regarded as a Maxwell uit i series with a Kelvi uit is capable of all the three basic viscoelastic patters. 88

3 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN Thus it imparts istataeous elastic respose because of the free sprig, viscous flow because of the free dashpot ad fially delayed elastic respose from the Kelvi uit. This study also focuses o five parameter model i which the module of elasticity ad viscosity coefficiets are assumed to be space depedet i.e. fuctios of x. Further, shearig strai i t a ad stress are take as harmoic fuctios of time t i.e. a ae ad a a e e The model is justified with the help of cyclic loadig for maxima it it. or miima. Here all the characteristics of the viscoelastic properties of the material, as determied uder the harmoic oscillatios, are the frequecy depedece of the compoets of the complex modulus or the frequecy depedece of the phase agle. Costitutive Relatio For Five Parameter Model The five parameter model cosists of two sprigs S( ), S( ) where ad are the modulli of elasticity associated to them ad three dash-pots D, D, D where 3 3, ad 3 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, ad u respectively specify the oly o-zero compoets of stress, shearig strai ad particle displacemet. The model has be divided ito three sectios, I, II, III. I fig., the sectio I, sectio II ad sectio III has oe sprig ( ) S, two dash-pots D, D respectively. oe sprig ( ) S ad oe dash-pot D 3 3 Figure- Five parameter viscoelastic model 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 a, a, a 3 be the three shearig 89

4 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN strais elogatios i respective sectios coected i series, the total elogatio is a a + a + a 3. 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., where ad are the stresses i the sub-sectios. Relatio for stress ad strai for D for sectio II (represeted by sigle dash-pot) is a () Sice the sub-sectio II is represeted by a Maxwell- elemet, the the relatio is expressed as D η Sice, for Sectio II, therefore = D a () D D D a η For sectio I, for the sprig S( ), the stress-strai relatio is give by (3) a (4) For Sectio III; for the dash-pot D3 3, the stress strai relatio is give by a 3 3 The Stress-strai relatio for the model represetig the viscoelastic body for total stress ( ) ad strai a ca be obtaied from Eq. (3), Eq. (4) ad Eq. (5) as: D D D Da 3 3 (6) (5) Now we take ij = ij = i = Si( i) j Dj( j ) (7) Where, Si( i) elastic modulus of sprig ad Dj( j ) = viscosity of dash-pot, ij = j, i ( i, ; j,,3) Usig, Eq. (6) ad Eq. (7), we get. D 3 D D D a 3 (8) 9

5 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN Put R, R 3, R 3 = R3., R 4 = R4 3 i Eq. (8), we get 3 4 D R R D R R D R D a D R D a (9) The Eq. (9) ca be writte i terms of differetial operator form as m Dax, t md x, t () m where, the order m ad of sums o right had side ad left had side i the Eq. () 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, = R3 R4, d D. Eq. () is the required differetial operator form dt of costitutive relatio for the model for viscoelastic material to be studied. Behavior of the model ca be discussed as I. O comparig the d order derivatives, we get a (a) Eq. (a) implies that the istataeous behavior is elastic due to sprig oly i II. sectio-of the model. O comparig Fist order derivatives, we get 3 a a (b) III. From Eq. (b), the behavior is elastic but effect of viscosity of dashpots, ad 3 is apparet due to the presece of relaxatio time ij. O comparig zero th order derivatives, we get 3. (c) The zero th order derivatives show that the model is viscoelastic i ature. Hece, from the above discussio, it is clear that at the start the higher order derivates of time domiate, but as the time passes their domiacy decreases. overig Equatios For Five Parameter Viscoelastic Model Oe of the goverig equatio for the viscoelastic model is costitutive relatio ad is (Lakes (998)) t,, tt t f,,,,, () (), tt, t, tt, t 9

6 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN The equatio of motio is The displacemet-strai relatio is u (3), x, tt a u,x (4) From Eq. (3) ad Eq. (4), we ca write log u a tt (5), x, xx, x, x, x, tt,, x Usig Eq. (), Eq. (3), Eq. (4) ad Eq. (5) the shearig stress field is +,t, ttt, tt = log + log, xxt, x, xt, xx, x, x (6) Solutio For Five Parameter Viscoelastic Model We assume that the solutio σ (x, t) of Eq. (6) may be represeted by the Friedlader Series (947) xt, F t hx. = F x, x t hx, With, F F, F, x h, x F, F, t F. =, (7) The various derivatives stress with respect to x ad t are F, F F F, t,, tt,, ttt 3, F h F, x, x, F h h F + h x F, xx, x, xx,, xxt F h, x h, xx F h, x F 3, xt F h, x F, (8) From Eq. (7) ad Eq. (8) h, x( log), x h, x h, x F F F ( ) log log F h F, x h, x F h, x log h, xx, x, 3, xx x h F, x 3 (9) Comparig the coefficiet of F, we get, log x Comparig the coefficiet of F, we get, log () x 9

7 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN = h ( ),,,, x log x h x h xx log, x Comparig the coefficiet of F, we get = h, x ( log), x h, x h, x h, xx Comparig the coefficiet of F 3, we get () () = h, x (3) By puttig the value of ad i Eq. (3), we get From Eq. () ad Eq. (), we get From Eq. () ad Eq. (5) h, x (4) h, x( log), x h, x h, xx (5) (6) By puttig the values of,,, i Eq. (7), we get Fially we get, ( R3 R4 ) =. R ( R + R ) R 3 4 (7) R R R R (8) Eq. (8) is the expressio for relaxatio time for five parameter viscoelastic model. Dyamic Loadig d Its Justificatio The time parameter t is itroduced ito a experimetal scheme i dyamic experimets by cyclic deformatio of the specime, frequecy of the oscillatios plays the role of the time factor. The cyclic deformatio is the fudametal process of determiig the mechaical 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 a, iduced i elastic body which ca be expressed by a harmoic actio as: aa iwt e (9) 93

8 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN Where, a is the amplitude, is the frequecy of oscillatios ad t is the time. ccordig to Hooke s law, stress is Where, = a, at t= iwt e (3) For a elastic body, the strai ad stress vary harmoically ad there is o lack i the harmoic motio i phase as both have iwt 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: a ie e iwt it Where, is the viscosity of the body ad a at t=. Thus, for a viscous (3) 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 zero ad 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, a a iwt e ad e i( wt ) If we represet the projectio of the stress vector o axis of co-ordiates by takig x ad y, where ad represet that real ad imagiary parts respectively. If the strai is iitially set harmoically, the the modulus of viscoelastic body with harmoic loadig ca be writte as The phase agle is give as * * = * a = * a i a a i (3) ta 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 3 3 (33) 94

9 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN 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 * iwt t ae (34) Where, a a iwt e Usig, the relatio a for a elastic body, the costitutive relatio for the physical state represetig the five parameter model is give by Eq. (9). Usig Eq. (3), Eq. (34) i Eq. (9), we get O solvig, we get i i R R i ( R R ) a * 3 4 * ( i ) = * ( i ) = iwt e = Ri a Ri R3 R4 R R i R ir R Ri 3 R4 i iwt e (35) R R R R3 R4 R R3 R4 R i Separate Eq. (38) ito real ad imagiary parts, we get = = d loss taget is give by R R R R R 3 4 R R R R3 4 R R3 R4 R R R R R3 R4 R ta R R R3 4 R R R R R To fid the values of, we put R R R 3 4,. 3 4 B R, E R R, C E, i Eq. (4), hece (36) (37) (38) (39) (4) (4) Where, Now by takig 3 B C E (4) 95

10 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN D ( ( i)) (43) d Where D dt With the help of Eq. (4) ad Eq. (43) we get B B E C 3 E B C B C E 3 E E B C C C B C 3 4 E E E B 4 C 3 BC C 3 B C 4 B C E 3 C E 3C C B B B B (44) Eq. (44) gives the dispersio equatio for wave propagatio. It is a cubic i, givig three roots, the either it has oe real root ad other complex roots as complex roots always occur i cojugate pairs or all the three roots are real ad at has either a maximal value or miimum value at these roots.. Therefore, takig roots as,,, we get Sum of roots, Product of roots take two at a time, Products of roots, 3 = E 3 C B 3 3 = 3 E 3C C B B C 3 = B To determie,, through Eq. (47) seems ot to be so easy, takig oe of the root for 3 (45) (46) (47) the extreme value of as (48) C. To fid the other two roots, for the Eq. (44) from Eq. (45), Eq. (46) ad Eq. (47), such that 3 3 = E 3 C B (49) 96

11 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN = C B (5) The Eq. givig, ca be expressed as 3 We get the roots, C = (5) B B x E 3 C x C B B B (5) x E C E C C, 3 E 3 C B B E 3 C B (53) B Whe 4C B E 3 C E B 3 BC B Takig ve sig, 4BC E B 3( BC) BE C = 3 C B C = 3 B BE BC C 3 (54) BE 3 BC B (55) Error due to approximatio is From, Eq. (46) pproximate value 3 C B C B 4 E 3 3 = 3 = 3 3 E BC C B C C C E 3 E B3 BC Error ca be calculated by subtractig Eq. (57) ad Eq. (58) B B B BC (56) (57) (58) 97

12 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN Error (ξ) = E B 3 - BC C C C E 3 B Takig the +ve sig, we get E B3 (ξ) = 3 BC C E BC C B B BC BC E B BC B B E B 3 B 3( ) CB E B 3 BC B E B 3( BC) 3 3 B 3 BC B E E B 3 BC C C (59) (6) (6) Case- t very low frequecies,, (from Eq. (4)) ( R R R R 3 4 ), The it is to be iferred that durig the cyclic loadig iitially ie.. R R3 R4 must be a poit of maxima or miima betwee ad. R Case- lso, t very high frequecies, (from Eq. (4)) ( R R ) (6), there 3 4 R3 R4 m (63) R R ( ) But for = R3 R4 it is observed that for C there must be a poit of maxima as whe iitially () icreases from zero to maximum value m ( R3 R4 ) R R (64) ad agai states R R3 R4 that dimiishig ad reaches zero at, which justifies the model for the R Viscoelastic materials. Whe relaxatio is applied to the model i.e. the model is uder the ifluece of costat deformatio, the specime represetig the model is deformed to the 98

13 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN give strai a ad after which it is maitaied costat, where as the stress required to maitaied these strais ( a ) dimiishes at a a costat, uder thermal coditios. The Costitutive equatio uder costat deformatio (strai) a a costat reduces to (65), tt, t Where,,, 3 =. 3 Eq. (65) ca be solved, we takig the roots of the auxiliary equatio as Where, ad are relaxatio times of the specime. m ad m. (66) m m (67) 3 mm. ( ) 3 From, Eq. (67) ad Eq. (68), the Eq. (65) becomes The Solutio of Eq. (69) is To elimiate ad D m m D mm ( ) mt t e e da t, t, Eq. (7) reduces to a, dt Hece, (68) (69) m t (7) (7) Where, m (7) m m, m m m m m m m m m mt mt m t m t t me me me me (73) 99

14 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN For sufficietly large time, t, so that. Hece with loger periods of observatio, the stress i the specime will drop to zero, i.e. equilibrium state will be achieved. Complex Viscosity Let, here ot the strai is specified but the stress which varies by the harmoic law: (74) i t e Let the model istatly respod to the chage i stress by udergoig a strai equal to I, which occur i phase with () t.where I is the istataeous compliace viscous flow develops by the law O itegratig, we get da () t dt (75) i t a e i () t i (76) Therefore, there exist third strai compoets, which is out of phase with the specified chage i stress. Now, the strai at () is give by i t a( t) I ( t) e i ( t) (77) where, is the amplitude of strai, which is out of phase with () t by the phase agle. Let I be the complex compliace as the iverse of complex modulus such that I, hece I at () () t a i or, I I i e a a I I cos i si I I I i I (78) Where I a a cos ; I si So it is clear that for I the quatity I correspods to the istataeous elastic deformatios of the material, ad correspods to viscous flow. Therefore, the viscoelastic behavior of 3

15 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN the material is govered by the values of I ad I. ssumig that I I ad I.Usig relatio I, the relatio betwee complex modulus ad complex compliace is obtaied as: I I ; I I I I ; I Usig Eq. (39) ad Eq. (4), we get the complex compliace for five parameter viscoelastic model. The taget of the agle (the loss factor) is expressed i terms of the ratio of the I compoets of the complex compliace as well as i terms of the ratio ta I I (79). Whe the siusoidally varyig stress is specified, the chage i the rate of strai ca be followed such that The we defie the complex viscosity da t (8) dt i a aie ia Where i e si icos i (8) a a i a a si, = cos a Here a is the amplitude of the strai rate. Sice () t at () where cos ; si a a Usig the above relatios, we get the relatio betwee complex viscosity ad complex modulus as: ; (8) The quatity is ofte for simplicity called just the dyamic viscosity. 3

16 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN Costitutive Relatio For Four Parameter Model We cosider the four parameter model with two sprigs S( ), S( ) ad two dash-pots D, D with viscoelasticity ad respectively (Fig.). The sprigs represet recoverable elastic respose ad dash pot represets elemets i the structure givig rise to viscous drag. Let a be the strai i S( ), a be the strai alog dashpot D ad a3 be the strai i the Kelvi model. Fig. represets the sketch of the stadard four parameter viscoelastic models. The stress v/s strai behavior for costat stress ( ) with time ( t a ) has bee show i fig.. Here,, are the modulli of elasticity,, are Newtoia viscosities coefficiets ad take as fuctios of x i the o-homogeeous case. Figure- Rheological model ad its respose The stress strai equatio for four parameter model is of geeral form. four parameter model cosistig of two sprigs ad two das-pots may be regarded as a Maxwell uit i series with a Kelvi uit as illustrated i Fig.. Let is stress ad a is shear strai; the relatios betwee them are give as a a a a, a, a, a, a (83) Elimiatig a, a, a 3, we get the costitutive Eq. () as a a (84) 3

17 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN Dyamic Respose Of Four Parameter Viscoelastic Model The followig values of strai are take for the dyamical respose of four parameter model From Eq. (84) ad Eq. (85), we get iwt * a e, a a e (85) iwt i * i O simplifyig Eq. (86), we get where, R R R R * (86) R8i R9 R5 R6 R7 i (87) * ca be writte i terms of real ad imagiary parts * 3 R9 R8 R5 R6 R7 i R8 R9 R8 R5 R6 R7 (88) The loss taget is, or * (89) R9 R8 R5 R6 R7, (9a) R9 R5 R6 R7 R R R R R R (9b) ta (9) From Eq. (9a), Eq. (9b) ad Eq. (9), we get Numerical alysis R9 R8 R5 R6 R7 R R R R R R ta The behavior of both the models have bee studied umerically as well as graphically, the rheological resposes are discussed by plottig a graph betwee for five parameter ad four parameter models. ad ad (9) verses 33

18 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN For five parameter model, we have Here, R R R B R, 3 4, ad R R 4 3. To calculate C R3 R4 3 B. C E, E R R, R, R 3, R. at differet frequecies, we assume the followig values Usig these values we get Now, 3 4.,.,.,., R 8.33, R 6.5, R 55, R , 84.75, B 8.33, C 9.945, E 4.83 where, R R R R3 R4 R 8.33, R 6.5, R 55, R , C E Fig. (3-4), has bee plotted for five parameter model. It is quite clear from graph, form fig. 3 ad fig. 4 for =7, there is peak for the graphs icreases both verse ad verses, as the value of ad decreases i.e the expoetial decay takes place. However, the decay i fig. 4 is steeper as compared i fig W Figure-3 Variatio of verses for five parameter model 34

19 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN W Figure-4 Variatio of verses for five parameter model Four parameter model, we have R9 R5 R6 R7 R R R R R R The values of the parameters for studyig the rheological respose are.,.,.,., R5 5, R6 5.5, R7, R8 5.5, R9 55. Now, graph betwee R9 R5 R6 R7 R9 R8 R5 R6 R7 verses is plotted by takig above equatio. The parameters take for. this case are the same. The fig. (5-6), shows that as the value of icreases both ad decreases, but the value of becomes costat after =55 for four parameter model W Figure-5 Variatio of verses for four parameter model 35

20 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN W Figure-6 Variatio of verses for four parameter model Coclusios It ca be cocluded that the both models possess a excellet potetial for proper represetatio of the time depedet behavior of a viscoelastic medium subjected to loadig ad uloadig. However, five parameter models are slightly better tha four parameter model. The viscous strais due to a costat stress are foud to icrease liearly with time for both the models. Moreover, after the removal of stress, the viscous strai is foud to remai costat with time. Durig the cyclic loadig iitially there must be a poit of maxima or 3 4 miima for five parameter model betwee ad. For sufficietly large relaxatio time for five parameter model, the stress i the specime will drop to zero. The phase shift agle for the viscoelastic body for five parameter model must be betwee zero ad. The use of five parameter ad four parameter models are mostly restricted i the field of rock mechaics. Thus, both models ca be used i determiig the time-depedet behavior of a viscoelastic medium. ckowledgemets The authors are thakful to the referees for their valuable commets. Refereces: lfrey T., No-homogeeous stress i viscoelastic media, Quarterly of pplied Mathematics,, 3, (944). Barbera J., ad Herrera J., Uiqueess theorems ad speed of propagatio of sigals i viscoelastic materials, rchive for Ratioal Mechaics ad alysis, 3, 73, (966). 36

21 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN chebach J. D., ad Reddy D. P., Note o the wave-propagatio i liear viscoelastic media, ZMP, 8, 4-43, (967). Bhattacharya S., ad Segupta P.R., Disturbaces i a geeral viscoelastic medium due to impulsive forces o a spherical cavity, erlads Beitr eophysik, Leipzig, 87:8, 57 6, (978). charya D. P., Roy, I., ad Biswas P. K., Vibratio of a ifiite ihomogeeous trasversely isotropic viscoelastic medium with a cylidrical hole, pplied Mathematics ad Mechaics, 9:3, -, (8). Bert C. W., ad Egle. D. M., Wave propagatio i a fiite legth bar with variable area of cross-sectio, J. ppl. Mech. (SME), 36, 98-99, (969). bd-lla. M., ad hmed S. M., Rayleigh waves i a orthotropic thermo-elastic medium uder gravity field ad iitial stress, J. Earth Moo Plaets, 75,85-97, (996). Batra R. C., Liear costitutive relatios i isotropic fiite elasticity, Joural of Elasticity, 5, 43-45, (998). Murayama S., ad Shibata T., Rheological properties of clays, 5 th Iteratioal Coferece of Soil Mechaics ad Foudatio Egieerig, Paris, Frace,, 69-73, (96). Schiffma R. L., Ladd C. C., ad Che. T. F., The secodary cosolidatio of clay, rheology ad soil mechaics, Proceedigs of the Iteratioal Uio of Theoretical ad pplied Mechaics Symposium, reoble, Berli, 73 33, (964). Jakowski R., Wilde K., ad Fujio Y., Poudig of superstructure segmets i isolated elevated bridge durig earthquakes, Earthquake Egieerig & Structural Dyamics 7, 487-5, (998). agostopoulos S.., Poudig of buildigs i series durig earthquakes, Earthquake Egieerig & Structural Dyamics, 6:3, , (988). Jakowski R., No-liear viscoelastic modellig of earthquake-iduced structural poudig, Earthquake Egieerig & Structural Dyamics, 34:6, 595-6, (5). Muthukumar S., ad Desroches R., Hertz cotact model with oliear dampig for poudig simulatio, Earthquake Egieerig & Structural Dyamics, 35, 8-88, (6). Westermo B.D., The dyamics of iterstructural coectio to prevet poudig, Earthquake Egieerig & Structural Dyamics, 8, , (989). Kakar R., Kaur K. ad upta K.C., alysis of five-parameter Viscoelastic model uder Dyamic Loadig, J. cad. Idus. Res., :7, 49-46, (). Kaur K., Kakar R. ad upta K.C., dyamic o-liear viscoelastic model, Iteratioal Joural of Egieerig Sciece ad Techology, 4:, , (). 37

22 Europea Scietific Joural pril 3 editio vol.9, No. ISSN: (Prit) e - ISSN Kaur K., Kakar R., Kakar S. ad upta K.C., pplicability of four parameter viscoelastic model for logitudial wave propagatio i o-homogeeous rods, Iteratioal Joural of Egieerig Sciece ad Techology, 5:, 75-9, (3). Kakar R., ad Kaur K., Mathematical aalysis of five parameter model o the propagatio of cylidrical shear waves i o-homogeeous viscoelastic media, Iteratioal Joural of Physical ad Mathematical Scieces, 4:, 45-5, (). Lakes R.S., Viscoelastic solids, CRC Press, New York, (998). Friedlader F.., Simple progressive solutios of the wave equatio, Proc. Camb. Phil. Soc., 43, 36-73, (947). 38