Applicability of Four Parameter Viscoelastic Model for Longitudinal Wave Propagation in Non-Homogeneous Rods

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1 Applicability of Fou Paamete Viscoelastic Model fo Logitudial Wave Popagatio i No-Homogeeous Rods KANWALJEET KAUR Faculty of Applied Scieces, BMSCE, Muktsa-56, Idia RAJNEESH KAKAR* Picipal, DIPS Polytechic College, Hoshiapu-46, Idia SHIKHA KAKAR Assistat Pofesso, SBBSIET, Jaladha, Pujab-44, Idia KISHAN CHAND GUPTA Faculty of Applied Scieces, BMSCE, Muktsa-56, Idia * of Coespodig Autho: kaka_63@ediffmail.com Abstact: This study is coceed to check the validity of a fou paamete viscoelastic model fo logitudial wave popagatig i the o-homogeeous viscoelastic ods of vayig desity. The ods ae assumed to be iitially ustessed ad at est. I this study, it is assumed that desity, igidity G ad viscosity of the specime i.e. od ae space depedet ad obey the laws e, G Ge 3 ad e. The method of o-liea patial diffeetial equatio (Eikoal equatio) has bee used fo fidig the dispesio equatio of logitudial waves i ods. A method fo teatig eflectio at the fee ed of the fiite ohomogeeous viscoelastic od is also peseted. All the thee cases take i this study ae discussed gaphically by usig MATLAB. Keywods: Logitudial Waves; Viscoelastic Media; Fiedlade Seies; Fou Paamete Viscoelastic Model.. Itoductio The viscoelasticity theoy is used i the field of solid mechaics, seismology, eploatio geophysics, acoustics ad egieeig. The solutios of may poblems elated with wave-popagatio fo homogeeous media ae available i may liteatues of cotiuum mechaics of solids. I eithe case the complete solutios to elastic o viscoelastic poblems ae kow whe mateial paametes ae teated idepedet of positio. I ecet yeas, howeve, sufficiet iteest has aise i the poblem coected with bodies whose mechaical popeties ae fuctios of space, i.e. o-homogeeous bodies. These poblems ae useful to udestad the popeties of polymeic mateials ad idustial elated applicatios. May eseaches Alfey [], Babea [], Achebach [3], Bhattachaya [4] ad Achaya [5] fomulated ad developed this theoy. Futhe, Bet [6], Abd-Alla [7], Bata [8] successfully applied this theoy to wavepopagatio i homogeeous, elastic media. Muayama [9] ad Schiffma et al. [] have poposed highe ode viscoelastic models of five ad seve paametes to epeset the soil behavio. Recetly, Kaka et al.; [, 3] ad Kau et al.; [] aalyzed vaious viscoelastic models ude Dyamic Loadig. I most of the liteatue, the poblems of o-homogeeity ae take as idepedet of space coodiate. But i this study, we coside the wave popagatio i o-homogeeous media, whe desity igidity G ad viscosity of the mateial ae space depedet such that the wave velocity is also space depedet. The poblem is solved with Eikoal equatio whe the wave equatio is appoimated usig WKB theoy. The displacemets assumed i the poblem ae so small that ude isothemal coditios, the liea costitutive laws hold. The displacemet ad stess epessios ae solved fo time depedet displacemet ad stess bouday coditios. The pape eds with umeical aalysis by takig mateial paametes.. Fomulatio of Poblem Let us coside wave is popagatig i oe dimesioal o-homogeeous semi-ifiite od, the ed of the od is kept at =. We coside the fou paamete model with two spigs S( G), S( G ) ad two dash-pots D, D with viscoelasticity ad espectively (Fig.). The spigs epeset ecoveable elastic espose ad dash pot epesets elemets i the stuctue givig ise to viscous dag. Hee G ad ISSN : Vol. 5 No. Jauay 3 75

2 Gae elastic paametes, ad ae viscoelastic paametes. Let be the stess ad a be the stai i the model. Let a be the stai i S( G ), a be the stai alog dashpot D ad a3 be the stai i the Kelvi model. Fig. epesets the sketch of the stadad fou paamete viscoelastic models. The stess v/s stai t ) has bee show i fig.. Hee, G, behavio fo costat stess ( ) with time ( a G ae the modulli of elasticity,, ae Newtoia viscosities coefficiets ad take as fuctios of i the o-homogeeous case. Fig. Rheological model ad its espose The stess-stai elatio fo the fou paamete viscoelastic models ae costituted by the equatios [8] Ga, a, Ga, a. 3 3 Elimiatig a, a, a 3 fom Eq. () we get the costitute equatio fo the fou paamete model: G G G GG GG Ga a () The stess stai equatio fo fou paamete model is of geeal fom: B B A a Aa (3) Whee Bi s ad Ai sae the coefficiets made up of combiatios of theg, G ad, ad deped upo the specific aagemet of the elemets i the model. I opeato fom the Eq. () ca be witte as t t t t B B A A a (4) The equatio of motio ad stai-displacemet elatio is give by U (5) t U a (6) Whee ( ) is the vaiable desity of the mateial. Diffeetiatig Eq.(5) w..t,we get u, tt Diffeetiatig Eq.(6) w..t. t,we get () (7) ISSN : Vol. 5 No. Jauay 3 76

3 a u t t Agai diffeetiatig w..t. t, a u u, tt u, tt (8) t t t Usig Eq. (7) ad Eq. (8), Eq. (3) gives,ttt, B, tt B, t A, t A log,, t A, A ( log),, (9) 3. Method of Solutio Let the solutio (, t) of Eq. (9) may be epeseted by the seies [4] t, AF th, A () Whee, F F (whee, =,, 3.) with Ft, F ad F, h, F () ad fo assume that A ad the deivatives of may be obtaied by tem-wise diffeetiatio of Eq. (), the pime i Eq. () deotes diffeetiatio with espect to the agumet coceed, ad by usig Eq. () ad Eq.() we elate all F s to F by successive itegatios. The Solutio of equatio Eq. (9) i the fom of Eq. () ca be obtaied by takig a phase fuctio h ( ), h ( ) satisfies the Eikoal equatio of geometical optics [5] dh( ) d G c Whee c = c() is the vaiable wave speed fo elastic logitudial waves i a medium whose modulus of elasticityg.usig, Eq.(), Eq.() ad the successive deivatives of, t w..t. t ad i equatio Eq.(9), we get AF G G G AF GG AF 3 G " " G AF AF h AF 3h AF h AF AF h GG AF " " AF h AF h AF h AF AF h GG O simplifyig the Eq. (3) usig Eq. (), we get the amplitude fuctio satisfy the equatio h A log h h"( ) A,(,, ), Q (4) Whee, G G G Q A" log, ( ), " A h log h h A h G G A" ( l og), A () (3) ISSN : Vol. 5 No. Jauay 3 77

4 Sice the wave is tavellig alog -ais, theefoe, itegatig Eq. (), we get h ds h (5) cs () Sice Eq. (4) is a fist ode liea diffeetial equatio i obtaied as [9] A.Theefoe the solutio of Eq. (4) ca be z l l ( ) Q l l s A A ep m s ds c s ep m z dz s ds (,, ) (6) Whee, l c ad m c. The plus sig is associated with wave tavelig i the positive diectio of ad the mius sig is associated with the waves tavellig i the egative diectio of. suddely applied at the ed of the od ad theeafte steadily Let a impulse of magitude maitaied, that is, t H( t) (7) Fom Eq. () ad Eq. (7), we have AF th H( t) (8) Thus we choose [5] if A if o h ad F H() t () The solutio of Eq. (9), fo the waves tavellig i the positive diectio of is geeated by bouday stess Eq. (9), is th t h ds (), t A H th A H t!! c( s) Whee, h Whee, ds () cs () A ae give ecusively by Eq. (6) (with uppe sigs) i combiatio with Eq. (9). The fist tem appoimatio leads to Eq. () as l ds, t ep msds H t l (3) c( s) The Eq. (3) epesets a tasiet stess wave which stats fom the ed with amplitude ad moves i the positive diectio of with velocity c(). Hece, it is modulated by the facto (9) ISSN : Vol. 5 No. Jauay 3 78

5 l ep msds l (4) Futhe tems i the appoimate solutio may be obtaied ecusively fom Eq. (8). The solutio of Eq. () applies util the wave movig i the positive diectio of stikes eithe a iteface (i the case of a composite od) o at ed (i the case of a fiite od). We will show that eflected waves ae poduced at the othe ed of the fiite od while both eflected ad tasmitted waves ae poduced at a iteface betwee two dissimila media. 4. Viscoelastic Model Applied to a Paticula Case Fo the sake of coceteess ad fo studyig the qualitative effect of o-homogeeity o the logitudial wave popagatio i o-homogeeous fou paamete viscoelastic ods, it is assumed that desity, igidity of the specime i.e. od ae space depedet ad obey the laws G ad viscosity 3 e, G Ge, e (5) If, 3 i.e. desity igidityviscosity (6) Case-I Whe, 3, the fom Eq. (5), we get e, G G e, e (7) Theefoe, fom Eikoal equatio of geometic optics e dh( ) d G G e G c o c G = costat. (8) (9) Sice, the epoetial vaiatio of modulus of igidity G ad desity is simila, theefoe soud speed is costat i.e. o-homogeeous has o effect o speed ad phase of the wave is give h. So it c becomes the case of semi o-homogeeous medium (a medium whe chaacteistics ae space depedet while the speed is idepedet of space vaiable). The amplitude fuctio A satisfies the equatio h A h h"( ) AQ, (,, ) (3) whee, G G G Q A" A h ( ) h h" A h (3) G G A " ( ) A As the amplitude fuctio is give by Eq.(6), Fo this case l G e ( ), m G ( ) m, md ( ) m (3) ISSN : Vol. 5 No. Jauay 3 79

6 Hece s A A e ep mds ce ep mdz Q sds (,, ) (33) Fo this case the value of fist tem appoimatio, the stess fuctio is give by, t e ep mdshth (34) The tasiet stess wave which stats fom the ed with amplitude ad moves with costat velocity c Case II G i the positive diectio of is modulated by the facto 3 i.e. desity igidity viscosity, the fom Eq. (5), we get 3 e, G G e, e Fom Eikoal equatio of geometic optics dh( ) e e d G G e G c e ep mds. (35) G Hee, c e (37) The amplitude fuctio A satisfies the equatio h A e h A ( 3) h( ) ( ) " Whee, Q Q (,, ) G 3 A" K A K h e h A K A" K A h 3 G (,, ) ad K e. G Ad Amplitude fuctio A ( ) is give by Eq.(6). Fo this case l ( ) G e = ( ) G 3 l ad m ( ) e m ( ) 3 ( ) md G 3 l ( ) ( ) l ( ) e l() l () e (36) (38) ISSN : Vol. 5 No. Jauay 3 8

7 Theefoe z ( ) l () Q l s A ( ) A () e ep m s ds c s ep m z dz s ds (39) Fo this case the value of fist tem appoimatio, the stess fuctio is give by G ( ) 3 t, e ep e H h 3 Whee, h ( ) e G t Eq.(4) gives a tasiet stess wave which stats fom the ed with amplitude ad moves with Velocity G i the positive diectio of is modulated by the facto is modulated the facto c e G (4) ( ) 3 e ep e Fo the case of a thi o-homogeeous fiite viscoelastic od The case of a thi o-homogeeous fiite viscoelastic od is take its legth is i the limit L. The paametes ae take as m l p q, G G, G G, At the ed,, G G, G G,, At the othe ed of the od L, m L, G G K, G G K, K, K Fom Eikoal equatio of geometic optics m dh( ) d G G G l p q (4) (4) h ( ) G m c c G m (43) ISSN : Vol. 5 No. Jauay 3 8

8 The amplitude fuctio A satisfies the equatio p q h A h( ) h" ( ) AQ (,, ) Whee Q A G A h lq k " G G G G lq mp h h A A" A Ad Amplitude fuctio A ( ) is give by Eq.(6). Fo this case l ( ) G m l ( ) m ( ) G Ad m m p q m ( ) m p m p md ( ) G m q m q Hece we ca wite z l ( l s m A ( ) A () ep m s ds c s) ep m z dz Q sds (44) (45) Fo this case the value of fist tem appoimatio, the stess fuctio is give by m p m m p G m q m q, t ep H t h (46) ISSN : Vol. 5 No. Jauay 3 8

9 The epessio (46) epesets a tasiet stess wave which stats fom the ed with amplitude ad moves with Velocity c G m i the positive diectio of is modulated by the facto m p m m p ep G m q m q The solutio applies util the wave movig i the positive diectio of stikes at the ed L ad we assume the ed as stess fee ad thee is o futhe popagatio of wave due to absece of media ad will be eflected. I othe wods we ca say a eflected wave is poduced at the ed L of the fiite od. This ca be epeseted as A ( ) F th ( ), A,fo. (48) Hee idicates fo eflective wave. The amplitude fuctio A ( ) stats at ed L L eplacig.the phase fuctio h ( ) L i Eq.(48) satisfy the Eq.(44).Its solutio is obtaied fo a eflective wave which ad moves i the evese diectio of is give by Eq.(45) with the lowe sigs, ad is obtaied fom the Eq.(5) with the mius sig ad eplacig.as it is metioed i Eq.(7), a tasiet stess wave is poduced which stats at the ed at time t ad advaces with the speed ( ) c.its asymptotic epesetatio is give by, Whee ad h A is give by Eq. (45) ad Eq. (9) ds cs () G m t h t A Hth! m G m (47) i the positive diectio of Whee the uppe sig i Eq. (45) epesets the wave advacig alog the positive diectio of.we deoted i i i this wave by ad coespodigly its amplitude ad phase fuctio by A ( ) ad h ( ), espectively. It also implies that it is the icidet at the stess fee ed L. The eflected wave poduced at ed L is justified by the bouday coditios at L satisfied by i. i At t h L the icidet wave has aived at the ed L, ad by applyig coditio of stess fee ed L we get i A ( L) A ( L),,,,... (5) (49) ISSN : Vol. 5 No. Jauay 3 83

10 i h L h L (5) By usig the values fom Eq. (45), Eq. (49) ad Eq. (4), the eflected wave may be completely detemied. s l () l L s m i A( ) A( L) ep m s ds c s ep m z dz Q sds L,,,... (5) ad h ( ) L Whe the solutio ds ds cs () cs () L (53) i will apply fo h L h ( L) t h ( ). aives at At time t h, The eflected wave ad the bouday coditios must be satisfied. Ay umbe of eflectios ca be teated i a staightfowad mae by epeatig the pocess. The fist tem appoimatio to the eflected wave is give by m L (, t) ep m( s) ds m( s) ds Hth ( ). (54) L Whee h ( ) is give by Eq. (53). Fom Eq. (5), futhe tems of this appoimate solutio may be obtaied. 5. Numeical Aalysis To see qualitative effect of o-homogeeity o the logitudial wave popagatio i o-homogeeous fou paamete viscoelastic ods, the vaious gaphs ae plotted betwee ad o liea as well o logaithmic scales. Figs (-5), have bee plotted fo semi- homogeeous medium at vaious values of i.e. = (,,,-,- ), it is assumed that desity, igidity G ad viscosity of the od ae space depedet such that e, k ke, e. The stess atio is calculated fom the equatio, t e ep md The paametes take ae O usig above values we get, e ep. s. Whee, k m.., k., k.,.5,.6 By cosideig the above values of paametes ad Eq. (55), the gaphs ae plotted betwee stess atio i.e. v/ s. (55) ISSN : Vol. 5 No. Jauay 3 84

11 V/S (ON LINEAR SCALE) Fig. (Takig,,,, V/S ( ON LOG SCALE) Fig. 3 (Takig,,,, V/S ( ON LOG SCALE) Fig. 4 (Takig,,,, ISSN : Vol. 5 No. Jauay 3 85

12 V/S (ON LOG SCALE) Fig. 5 (Takig,,,, Figs (6-9) ae daw fo o- homogeeous medium, it is assumed that desity, igidity G ad viscosity 3 of the od ae space depedet such that, e k k e, e. The stess atio calculated fom the equatio of wave fot, give by k ( ) 3, t e ep e H th 3 k ( ) 3 e ep e 3 The paametes ae take as.,.5, 3.. Gaphs ae plotted betwee v/ s ae as show i figs (6-9) is (56).5 V/S (ON LINEAR SCALE) Fig. 6 (Takig 3.,.5,. ISSN : Vol. 5 No. Jauay 3 86

13 .5 V/S ( ON LOG SCALE) ,.5,. Fig. 7 (Takig 3 V/S ( ON LOG SCALE) Fig. 8 (Takig 3.,.5,. V/S (ON LOG SCALE) Fig. 9 (Takig 3.,.5,. ISSN : Vol. 5 No. Jauay 3 87

14 Fo the case of fiite od, figs (-3) ae daw by takig the paametes as m.9,.8, p.7, q.6, K.5, L 3. Gaphs ae plotted betwee the equatio of wave fot, give by v/ s ae as show i figs (-3). The stess atio is calculated fom, t ep H t h m p m m p k m q m q m p m m p ep k m q m q (57) 6.65 V/S (ON LINEAR SCALE) Fig. (Takig m.9,.8, p.7, q.6, K.5, L 3. ISSN : Vol. 5 No. Jauay 3 88

15 6.65 V/S ( ON LOG SCALE) Fig. (Takig m.9,.8, p.7, q.6, K.5, L 3. V/S ( ON LOG SCALE) Fig. (Takig m.9,.8, p.7, q.6, K.5, L 3. V/S (ON LOG SCALE) Fig. 3 (Takig m.9,.8, p.7, q.6, K.5, L 3. ISSN : Vol. 5 No. Jauay 3 89

16 6. Coclusios. Whe the desity, igidity ad viscosity all ae equal fo the fist mateial specime, the soud speed is costat i.e. o-homogeeous has o effect o speed ad phase of the wave is give h. c So it becomes the case of semi o-homogeeous medium (a medium whe chaacteistics ae space depedet while the speed is idepedet of space vaiable). The logitudial speed will be equal to G c. Whe the desity, igidity ad viscosity ae ot equal fo the secod mateial specime, the speed of G soud vaies epoetial as c e 3. The case of a thi o-homogeeous fiite viscoelastic od is solved by takig o-homogeeous paametes ad the logitudial speed fo this paticula case comes out to be c G m Ackowledgemets The authos covey thei sicee thaks to DIPS Polytechic College ad BMSCE College fo facilitatig us with best facility. The authos ae also thakful to the efeees fo thei valuable commets. Refeeces [] Alfey, T. (944): No-Homogeeous Stess i Viscoelastic Media, Quat. Applied Math,, pp. 3. [] Babea, J.; Heea, J. (966): Uiqueess Theoems ad Speed of Popagatio of Sigals i Viscoelastic Mateials, Ach. Rat. Mech. Aal., 3, pp. 73. [3] Achebach, J. D.; Reddy, D. P. (967): Note o the Wave-Popagatio i Liea Viscoelastic Media, ZAMP, 8, pp [4] Bhattachaya, S.; Segupta, P.R. (978):Distubaces i a Geeal Viscoelastic Medium due to Impulsive Foces o a Spheical Cavity, Gelads Beit Geophysik, Leipzig, 87(8), pp [5] Achaya, D. P.; Roy, I.; Biswas, P. K. (8): Vibatio of a Ifiite Ihomogeeous Tasvesely Isotopic Viscoelastic Medium with a Cylidical Hole, Applied Mathematics ad Mechaics (Eg. Ed.), 9(3), pp.. [6] Bet, C. W.; Egle., D. M. (969): Wave Popagatio i a Fiite Legth Ba with Vaiable Aea of Coss-sectio, J. Appl. Mech. (ASME), 36, pp [7] Abd-Alla, A. M.; Ahmed, S. M. (996): Rayleigh Waves i a Othotopic Themo-elastic Medium ude Gavity Field ad Iitial Stess, J. Eath, Moo Plaets, 75, pp [8] Bata, R. C., (998): Liea Costitutive Relatios i Isotopic Fiite Elasticity, J. Elasticity, 5, pp [9] Muayama, S.; Shibata, T. (96): Rheological popeties of clays, 5 th Iteatioal Cofeece of Soil Mechaics ad Foudatio Egieeig, Pais, Face,, pp [] Schiffma, R.L.; Ladd, C.C.; Che, A.T.F., (964): The secoday cosolidatio of clay, Rheology ad Soil Mechaics, Poceedigs of the Iteatioal Uio of Theoetical ad Applied Mechaics Symposium, Geoble, Beli, pp [] Kaka, R.; Kau, K.; Gupta K.C., (): Aalysis of Five-Paamete Viscoelastic model ude Dyamic Loadig, J. Acad. Idus. Res., (7), pp [] Kau, K.; Kaka, R.; Gupta, K.C., (): A dyamic o-liea viscoelastic model, Iteatioal Joual of Egieeig Sciece ad Techology, 4 (), pp [3] Kaka, R.; Kau, K., (3): Mathematical aalysis of five paamete model o the popagatio of cylidical shea waves i ohomogeeous viscoelastic media, Iteatioal Joual of Physical ad Mathematical Scieces, 4(), pp [4] Fiedlade, F. G. (947): Simple pogessive solutios of the wave equatio, Poc. Camb. Phil. Soc., 43, pp [5] Kal, F. C.; Kelle, J. B. (959): Elastic waves popagatio i homogeeous ad ihomogeeous media. Joual of Acoustical Society Ameica, 3, pp [6] Moodie, T. B. (973): O the popagatio, eflectio ad tasmissio of tasiet cylidical shea waves i o-homogeeous foupaamete viscoelastic media, Bull. Aust. Math. Society, 8, pp [7] Caslaw, H. S.; Jaege, J. C. (963). Opeatioal Methods i Applied Math, Secod Ed., Dove Pub, New Yok. [8] Blad, D. R. (96). Theoy of Liea Viscoelasticity, Pegamo Pess, Ofod. [9] Gudasha, S.; Avta, S. (98). Popagatio, eflectio ad tasmissio of logitudial waves i o-homogeeous five paamete viscoelastic ods, (9), pp ISSN : Vol. 5 No. Jauay 3 9