One Dimensional State Space Approach to Thermoelastic Interactions with Viscosity

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1 7 IJSRST Volum 3 Issu 8 Print ISSN: Onlin ISSN: 395-6X Thmd Sction: Scincand Tchnology On Dimnsional Stat Spac Approach to Thrmolastic Intractions with Viscosity Kavita Jain Rnu Yadav Dpartmnt of Mathmatics Guru Jambhshwar Univrsity of Scinc and Tchnology Hisar Haryana India Dpartmnt of Mathmatics Govrnmnt Collg Adampur Hisar Haryana India ABSTRACT Prsnt work is concrnd with th solution of a on dimnsional problm in gnralizd thrmoviscolastic mdium with fractional ordr strain. Th formulation is applid in th contxt of Grn-Naghdi thory of thrmolasticity with nrgy dissipation. Stat spac approach togthr with Laplac transform tchniqu is adoptd to obtain th gnral solution. Numrical invrsion tchniqu is usd to driv th xprssions of diffrnt fild variabls in th physical domain. Numrical rsults ar givn and illustratd graphically. Kywords : GN thory Viscosity Fractional Ordr Strain. I. INTRODUCTION Thrmolastic wav propagation is of much importanc in diffrnt filds such as arthquak nginring nuclar ractors aronautics and astronautics tc. Th coupling btwn thrmal and strain filds gav ris to th coupld thory of thrmolasticity. Th thoris of gnralizd thrmolasticity wr dvlopd to amnd th classical thrmolasticity thory. By providing sufficint basic modifications to th constitutiv quations Grn and Naghdi [-3] producd a thory which was dividd into thr diffrnt parts rfrrd to as G-N thory of typ I II and III. Typ I is sam as classical hat conduction thory (basd on Fourirs law of hat conduction). Typ II prdicts th finit spd of hat propagation involving no nrgy dissipation. In typ III constitutiv quations ar drivd by including thrmal displacmnt gradint in addition to tmpratur gradint among constitutiv variabls. In th last fw yars fractional calculus thory has bn mployd succssfully in thoris of thrmolasticity and svral modls of fractional ordr gnralizd thrmolasticity ar stablishd by many authors. Shrif t al. [4] introducd th fractional ordr thory of thrmolasticity by using th mthodology of fractional calculus provd uniqunss thorm and drivd variational principl and rciprocity thorm. Ezzat [5] constructd a nw mathmatical modl of fractional hat conduction law in which th gnralizd Fourirs law of hat conduction is modifid by using th nw Taylors sris xpansion of tim fractional ordr dvlopd by Jumari [6]. Rcntly Youssf [7] drivd a nw thory of thrmolasticity with fractional ordr strain which is considrd as a nw modification to Duhaml- Numanns strss-strain rlation. In this papr th author postulatd a nw unifid systm of quations that govrn svn diffrnt modls of thrmolasticity in th contxt of on-tmpratur and two-tmpratur and on dimnsional problm for an isotropic and homognous lastic half-spac. This invstigation studis th on dimnsional problm of linar isotropic solid in thrmoviscolastic mdium subjctd to mchanical load. Th application of th prsnt work can not b ruld out in gophysics and arthquak nginring du to th importanc of thrmoviscolastic proprtis. Stat spac approach is mployd for th gnral solution of th problm. Th variations of th considrd fild variabls with th distanc ar prsntd graphically. σ ij ij u i (i = xyz) θ = T T T II. NOMENCLATURE Componnts of strss tnsor Componnts of strain tnsor Componnts of displacmnt vctor Tmpratur Absolut tmpratur IJSRST73853 Rcivd: Dc7 Accptd : 3Dc 7 Novmbr-Dcmbr-7[(3)8: 67-73] 67

2 T Th tmpratur of mdium in its natural stat assumd to b Dnsity of mdium Mchanical rlaxation tim Fractional strain paramtr t t t 3 t 3 t Lam s lastic constants Viscolastic rlaxation tims t Cofficint of linar thrmal xpansion c E k K c E Spcific hat at constant strain Thrmal conductivity 4 Matrial charactristic of GN thory III. BASIC EQUATIONS Th constitutiv rlation and govrning quations for a gnralizd viscothrmolastic problm undr th purviw of GN III thory with fractional ordr strain ar: ( D ) ( D ) () ij t ij t kk ij ij ij ( ui j u j i ) () u (3) ji j ( K ) ii k ii ce T Dt (4) i Hr a dot ovr a variabl dnots drivativ with rspct to tim t a comma rfrs to a spatial drivativ and th tnsor convntion of summing ovr rpatd indics is usd. IV. PROBLEM FORMULATION In th considration of on dimnsional problm th occupid rgion is < x < whos stat dpnds only on th spac variabl x and tim t. So th displacmnt vctor u and tmpratur θ can b xprssd in th following form u x = u(xt) u y = u z = θ = θ(xt). (5) Th govrning quations ()-(4) in on-dimnsional cas assum th shap xx ( )( Dt ) (6) u xx (7) x u x (8) 3 K k c E T( Dt ). (9) x tx Equation (8) can b xprssd as () Procding with th analysis w introduc dimnsionlss variabls dfind by th xprssions: ce x x u u c T T ( t ) ( t ) T whr and. () Substituting ths non dimnsional valus in quations (6) (9) and () w gt following non dimnsional quations (supprssing th prims): 68

3 ( Dt ) t t () L ( s)( s ) M s. (8) ( Dt ) t x t t t (3) x t (4) V. STATE SPACE FORMULATION Having chosn th tmpratur and strss componnt as stat variabls quations (7) may b prsntd in matrix form as whr k c K K T ar th coupling paramtrs. Using th Laplac transformation dfind as whr D V (xs) = A(s)V (xs) (9).() (5) ovr th quations ()-(4) and using th homognous initial conditions w gt th following quations ( s)( s ) ( s) ( s ) D s ( s)( s ) s D s (6) whr. Eliminating from (6) w arriv at th following systm of diffrntial quations Th formal solution of th diffrntial quation (9) may b writtn as V ( x s) xp[ A( s) x] V ( s) () whr and I is an idntity matrix of scond ordr. Th trms containing xponnts of growing natur in th spac variabl x hav bn discardd du to th rgularity condition at infinity. Th charactristic quation of matrix A(s) is obtaind as λ (L + M )λ + L M L M = () () D L L D M M whr M s ( s) ( s)( s ) (7) whr th roots λ λ of quation (3) must satisfy (3) L M L M L M. (4). Th Taylor sris xpansion of th matrix xponntial has th form. (5) M s ( s)( s ) L s s ( s)( s ) M Making us of th wll-known Cayly-Hamilton thorm w can xprss A and highr ordrs of th matrix A in trms of I and A. Thus th infinit sris in (5) can b truncatd as 69

4 xp[ A( s) x] a ( x s) I a ( x s) A (6) whra and a ar constants dpnding on x and s. Again by Cayly-Hamilton thorm th charactristic roots λ and λ of th matrix A must satisfy quation (6). Thrfor w hav. (7) On solving th abov linar systm of quations w obtain (8) Substituting th valus of a and a along with I and A into quation (6) w hav xp[ A( s) x] ( x s)( i j ). (9) ij whr th componnts Γ ij (xs) ar givn by x L L (3) ( ) ] x ( x s) [( M M ) x M M (33) ( ) ]. VI. APPLICATION W considr a homognous isotropic viscolastic mdium occupying th rgion x with quiscnt initial stat and boundary conditions in th following forms:. Mchanical boundary condition W will suppos that th mdium is subjctd to a mchanical shock at x = as follows: σ(t) = σ = σ H(t) (34) whrσ is a constant. By applying Laplac transform dfind in (5) w obtain. (35). Thrmal boundary condition Th mdium at x = is kpt at rfrnc tmpratur T i.. θ(t) = θ =. (36) Oprating Laplac transform on th abov quation on can obtain. Hnc solution () can b writtn as (3) V ( x s) V ( s). (3) ij s Plugging th valus of V ( x s ) and A() s I v into (3) and aftr som straightforward calculation th xprssions for conductiv tmpratur and strss ar valuatd as x ( x s) [( L L ) θ(s) = θ =. (37) Hnc w can utiliz th valus of σ and θ from (35) and (37) in (3) and (33) to finally achiv th solutions in th Laplac transform domain as s x x [ L ( )] (38) [( ) x x M ( M ) ]. (39) s Using dimnsionlss variabls and Laplac transform in (8) th displacmnt componnt may b valuatd as (4) 7

5 Tmpratur Displacmnt whr c T. Substitution of σ from (39) into th abov quation yilds u M x [ 3 ( ) ( ) s (4) x ( M) ]. VIII. NUMERICAL RESULTS AND DISCUSSION With an aim to illustrat th contribution of fractional strain paramtr mchanical rlaxation tim and viscosity cofficints and hat sourc on fild quantitis a numrical analysis is carrid out. For this purpos w hav takn th following valus of rlvant paramtrs: VII. NUMERICAL INVERSION OF THE TRANSFORM Th quations (38) (39) and (4) provid th xprssions for tmpratur strss and displacmnt in Laplac transform domain. To dtrmin ths in physical domain Laplac invrsion is applid with th hlp of numrical tchniqu basd on Fourir xpansion of functions prformd by Honig and Hirds [8]. Lt f (s) b th Laplac transform of function f(t). Th invrsion formula of Laplac transform stats that WV NV whr d is an arbitrary ral numbr gratr than all th ral parts of singularitis of f (s). Taking s = d + ιy and using Fourir sris in th intrval [T] w gt th approximat formula whr for t T (4) Figur: Effct of viscosity on displacmnt WV NV. (43) and N is a sufficintly larg intgr rprsnting th numbr of trms in th truncatd Fourir sris chosn such that int dt T ik R f d T (44) whr is a prscribd small positiv valu that corrsponds to th dgr of accuracy to b achivd Figur: Effct of viscosity on tmpratur 7

6 Displacmnt Strss Strss Tmpratur WV NV.8.6 β=. β=.5 β= Figur3: Effct of viscosity on strss In figurs -3 w hav studid th ffct of viscosity paramtr on diffrnt filds in thrmolastic mdium undr th cas with viscosity (WV) and no viscosity (NV). Fig. is drawn to obsrv th ffct of viscosity on displacmnt. Viscosity has dcrasing ffct on displacmnt in th rgion x.3 whil incrasing ffct aftr x =.3. Viscosity paramtr xhibits an incrasing ffct on tmpratur and strss filds which is clar from Fig. and 3 rspctivly. Initially tmpratur is zro which is in accordanc with th boundary conditions. Ultimatly all th curvs tnd to zro which is physically admissibl Figur5: Effct of fractional paramtr on tmpratur.45 β=..4 β=.5 β= β=. 3.5 β=.5 β= Figur4: Effct of fractional paramtr on displacmnt Figur6: Effct of fractional paramtr on strss Figurs 4-6 ar plottd to show th ffct of fractional paramtr β on displacmnt tmpratur and strss filds rspctivly. Numrically displacmnt incrass for x.5 whil dcrass for x >.5 with th incrmnt of valus of β untill it bcoms zro. It is noticd from Fig. 4 that th profil of tmpratur is almost sam for β =.5 and.8 xcpt th rgion.5 x 3 and hr tmpratur is largr in th cas of β =.5 as compard to th cas of β =.8. Numrical valus of tmpratur ar largr for β =. among all th considrd valus of β. Fractional paramtr xhibits dcrasing ffct on strss which can b varifid from Fig. 6. 7

7 IX. CONCLUSIONS According to abov analysis w can conclud th follwing points: [8]. Honig G. and Hirds U. (984). "A mthod for th numrical invrsion of Laplac transforms" J. Comput. Appl. Math All th filds ar rstrictd in a limitd rgion which is in accordanc with th notion of gnralizd thrmolasticity thory.. Viscosity paramtr has incrasing ffct on th considrd filds xcpt displacmnt (hr viscosity shows mixd ffct). 3. Th ffct of fractional paramtr on all th studid filds is vry much significant. Th rsults prsntd in this papr should prov usful for rsarchrs in matrial scinc and dsignrs of nw matrials. X. REFERENCES []. Grn A.E. and Naghdi P.M. (99). "A rxamination of th basic postulats of thrmomchanics" Proc. Roy. Soc. Lond. A []. Grn A.E. and Naghdi P.M. (99) "On undampd hat wavs in an lastic solid" J. Thrm. Strss [3]. Grn A.E. and Naghdi P.M. (993) "Thrmolasticity without nrgy dissipation" J. Elast [4]. Shrif H. El-Sayd A.M.A. and El-Latif A.M.A. (). "Fractional ordr thory of thrmolasticity". Int. J. Solids Struct [5]. Ezzat M.A. () "Thrmolctric MHD non- Nwtonian fluid with fractional drivativ hat transfr" Physica B [6]. Jumari G. () "Drivation and solutions of som fractional BlackSchols quations in coarsgraind spac and tim. Application to Mrton s optimal portfolio" Comput. Math. Appl [7]. Youssf H.M. (6) "Thory of gnralizd thrmolasticity with fractional ordr strain" J. Vib. Cont