STUDY OF STABILITY MATTER PROBLEM IN MICROPOLAR GENERALISED THERMOELASTIC

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1 STUDY OF STABILITY MATTER PROBLEM IN MICROPOLAR GENERALISED THERMOELASTIC arinder ingh kounil c.t. intitute of technology jalandhar india dr. gurinder ingh ara Lyallur khala college jalandhar india Abtract The theory of icroolar theroelaticity ha any alication. One for of the recent year concerning the roble of roagation of theral wave at finite eed and the oibility of econd ound effect etablihed a new thero echanical theory of deforable edia that ue a general entroy balance a otulated and the theory i illutrated in detail in the context of flow of heat in a rigid olid with articular reference to the roagation of theral wave at finite eed. Then theory of theroelaticity for non-olar bodie baed on the new rocedure wa dicued and eloyed the eigen value aroach to tudy the effect of rotation and relaxation tie in two dienional roble of generalied theroelaticity. Recently invetigation how the dynaic reone of a hoogeneou iotroic generalied theroelatic halface with void ubjected to noral tangential force and theral tre. In thi aer we introduce the eigen value aroach following Lalace and Fourier tranforation ha been eloyed to find the general olution of the field euation in a icroolar generalied theroelatic ediu for lane train roble. An alication of an infinite ace with an iulive echanical ource ha been taken to illutrate the utility of the aroach. The integral tranforation ha been inverted by uing a nuerical inverion techniue to get reult in hyical doain. The reult in the for of noral dilaceent noral force tre tangential force tre tangential coule tre and teerature field coonent have been obtained nuerically and illutrated grahically. Secial cae of a theroelatic olid ha alo been deduced. Keyword: Eigen value; Mechanical ource ; Lalace and Fourier tranfor ; Concentrated force; Roberg integration. Introduction The theory of icroolartheroelaticity ha been a ubject of intenive tudy. A corehenive review of work on the ubject wa given by Eringen (97) and Nowacki (9). There ha been very uch written in recent year concerning the roble of roagation of theral wave at finite eed. A generalied theory of linear icroolartheroelaticity that adit the oibility of econd ound effect wa etablihed by Bochi (99). Recently Green and Naghdi (99) etablihed a new theroechanical theory of deforable edia that ue a general entroy balance a otulated by Green and Naghdi (977). The theory i illutrated in detail in the context of flow of heat in a rigid olid with articular reference to the roagation of theral wave at finite eed. A theory of theroelaticity for non-olar bodie baed on the new rocedure wa dicued by Green and Naghdi (99).Bakhi Bera and Debnath () eloyed the eigen value aroach to tudy the effect of rotation and relaxation tie in two dienional roble of generalied theroelaticity. Kuar and Rani () tudied the deforation due to echanical and theral ource in generalied orthorhotictheroelatic aterial. Kuar and Rani () invetigated the dynaic reone of a hoogeneou iotroic generalied theroelatic halface with void ubjected to noral tangential force and theral tre. The icroolar theory wa extended to include theral effect by Nowacki (9) and Eringen (97).

2 Kuar and Chadha (98) derived the exreion for dilaceent icrorotation force tre coule tre and firt oent for a half - ace ubjected to an arbitrary teerature field and a articular cae of line heat ource ha been dicued in detail. The uniuene of the olution of oe boundary value roble of the linear icroolartheroelaticity wa invetigated by Craciu (99). Paarella (99) olved the initial-boundary value roble for icroolartheroelaticity and roved a uniuene theore for the roble. Mahalanabi and Manna (997) dicued eigen value aroach to linear icroolartheroelaticity by arranging baic euation of elaticity in the for of atrix deferential euation in the Hankel tranfor and extended the aroach to linear theroelaticity. Marin and Luu (998) invetigated haronic vibration in theroelaticity of icroolar bodie. Kuar and Dewal () dicued the diturbance due to echanical and theral ource in hoogeneou iotroic icroolar generalied theroelatic half-ace.. Forulation and olution of the Proble We conider a hoogeneou iotroic icroolar generalied theroelatic olid in an unditurbed tate and initially at unifor teerature. We take a carteian yte (xy) and - axi ointing vertically into the ediu. Following Eringen (98) Lord and Shulan (97) and Green and Linday (97) the field euation and the contitutive relation in icroolar generalied theroelatic olid without body force body coule and heat ource can be written a. ν Tρ. Kρj!" # T $ %ν $# Ξ $ % ij αφ δ β φ γφ r r ij i j j i t ij λu T ( u u ) K ( u ε φ )- ν T τ r rδ ij µ i j j i j i ijr r δ ij t For the L-S (Lord Shulan) theoryτ Ξ and for G - L (Green Linday) theoryτ Ξ The theral relaxation τ and τ atify the ineuality τ τ > for the G-L theory only. However it ha been roved by Sturnin () that the ineualitie are not andatory for τ and τ to follow. For two dienional lane train roble arallel tox-lane we aue ( ( -

3 The dilaceent coonent u u and icrorotation coonent - deend uon x and t and are indeendent of co-ordinate y o that y. With thee conideration and uing (.) and introducing the non-dienional uantitie a C x x ω C ω T T T T C u u ν ρω T C ν ω ( ) K C µ λ ω. C λ µ ρ Now alying Lalace and Fourier tranfor defined by ( ) ( ) ex(-t)dt t x f x f ( ) ( ) ) e(- - dx x x f f ι on the et of euation (.)-(.) after ureing rie we get ( ) ( ) T d d d du u d u d τ ι φ ι ( ) T d d u d du d u d φ ι ι ( ) φ ι φ u d u d d d

4 d T d u ε d d { T } ( ) τ Ξ ι u ( τ ) λ µ K ρ C λ µ ρ C K ρ C KC ρω µ 7 8 ρ C ρ C λ T β ε ρ K ω. Euation (.9) - (.) can be written in the vector atrix differential euation for a d W d ( ) A( ) W ( ) W U O I DU A A A A f f f f f f A g g g g g g g g and O i the Null atrix of order with f l f f f ε ( τ Ξ)

5 g ( ) g l Ξ g ( τ ) ( τ ) g lε To olve the euation (.) we take W( ) X ( ) e for oe So we obtain A( )W( ) W( ) Thi lead to Eigen value roble. The characteritic euation correonding to the atrix A i given by which on exanion rovide u 8 σ σ σ σ./ι σ g g g g f f f f f f σ g g g g g g g g g g g g f f g f f g f f g f f f f g g f f g f f g f f g f f g f f g g g g f f σ g g g g g g g g g g g g g g g g g g f f g g σ f f g g f f g g f f g g f f g g f f g g f f g g f f - gg ggg - ggg. g ggg gggg gggg gggg The eigen value of the atrix A are the characteritic root of the euation (.9). The vector ( ) X correonding to the eigen value can be deterined by olving the hoogeneou euation [ I] X ( ) A The et of eigen vector X ( ) ;... 8 ay be defined a

6 ) ( X ) ( X ) ( X a b X ( ) - c a b X ( ) - c l -a b X ( ) - c l a -b X ( ) -c ( ) { } ( ) ( )( )( ) a }] { [ Ξ τ τ ε τ ( ) { } ( )( ) ( )( )( ) ] } { [ Ξ b τ τ ε τ ι ( )( ) ( ) { } [ ] b a c τ ι τ ε Ξ ( ) { } ( ) ( )( )( ) ] } { [ τ Ξ τ ε τ Thu olution of euation (.) i a given by Shara and Chand.(99) ( ) ( ) [ ] e X E e X E ) W( - 7 E E E E E E E and 8 E are eight arbitrary contant. The euation (.) rereent a general olution of the lane train roble for iotroic icroolar generalied theroelatic olid and give the dilaceent icrorotation and teerature field in the tranfored doain..alication

7 Mechanical Source We conider an infinite icroolar generalied theroelatic ace in which a concentrated force where F i the agnitude of the force F F δ (x) δ (t) acting in the direction of the - axi at the origin of the Carteian co-ordinate yte a hown in fig.. The boundary condition for reent roble on the lane are MICROPOLAR THERMOELASTIC MEDIUM-II < F O x ν > MICROPOLAR THERMOELASTIC MEDIUM-I u(x t) - u(x t) u (x t) - u (x t) (x φ t) - φ(x t) T(x t) -T(x t) T T ( x t) ( x t) t ( x t) t ( x t) t ( t) t ( x t) F δ (x) δ (t) ( x t) ( x t) x F Making ue of euation (.)-(.7) and F in euation (.)-(.) we get the tree in K the non-dienional for with rie. After ureing the rie we aly Lalace and Fourier tranfor defined by euation (.8) on the reulting euation and fro euation (.) we get tranfored coonent of dilaceent icrorotation teerature field tangential force tre noral force tre and tangential coule tre for > are given by { } e e e e u % ( ) a E a E a E a E 7 8 u % ( ) b E e b E e b E e b E e 7 8 { } e e e e % ϕ ( ) - E E E E 7 8 T% ( ) c E e c E e c E e c E e 7 8

8 ( ) e ( ) ( ι ) e ( ι ) % t ( ) a ι b E a ι b E e 7 a b E a b E e ( ( ) ) ι ( τ ) ι ( τ ) ι a ( τ ) % t ( ) [ ι a b c τ E e 8 ( 8a b c ) E e ( 8a b c ) E7e ( 8 b c ) E 8e { } % ( ) E e E e E e E e for < the above exreion get uitably odified e.g. u % ( ) a E e a E e a E e a E e Making ue of the tranfored dilaceent icrorotation icrotretch and tree given by (.)-(.) in the tranfored boundary condition we obtain eight linear relation between the E i which on olving give F E E c a a c a a c a a ( ) ( ) ( ) F E E c a a c a a c a a ( ) ( ) ( ) F E E c a a c a a c a a ( ) ( ) ( ) 7 F E E c a a c a a c a a ( ) ( ) ( ) 8 {( ) ( ) ( )} c {( ab ab ) ( ab ab ) ( ab ab )} c {( ab ab ) ( ab ab ) ( ab ab )} c ( a b a b ) ( a b a b ) ( a b a b ) [ c a b a b a b a b a b a b { } Thu function u% u% % φ T % t% t% and % have been deterined in the tranfored doain and thee enable u to find the dilaceent icrorotation teerature field and tree. ]

9 Cae I : For L-S theory a b and c in the exreion (.)-(.) take the for a [ { { ( τ ) } ( ) ( )( ) ε τ } ] ι b [ { } { ( τ ) } ( )( ) ( )( ) ε τ ] c ( ι ) ( τ ) ε a b { } [ { ( τ ) } { ( ) } ( )( ) ε τ ] ; and ± ( ) are root of the euation (.9) in which σ σ σ and σ are obtained reectively fro exreion (.)-(.) by taking τ Ξ. Cae II : For G-L theory a b and c' in the exreion (.)-(.) take the for a [ { { ( τ ) } ( ) ( )( ) ε τ } ] ι b [ { } { ( τ ) } ( )( ) ( )( ) ε τ ] c ( ι ) ( τ ) ε a b { }

10 [ { ( τ ) } { ( ) } and ( )( ) ε τ ] ; ± ( ) are root of the euation (.9) in which σ σ σ and reectively fro exreion (.) - (.) by taking Ξ σ are obtained Cae III : For Green and Naghdi theory (G-N) euation (.) (.) and (.) can be written a. ν Tρ t K TρC T t. $ t t i j λ u r r δ µ i j ( u u ) K ( u ε φ ) i j ji ji i j r r νtδ and K i not the uual theral conductivity but a aterial characteritic contant in G - N & theory and i given K C ( λ µ ) With the hel of euation (.)-(.8) and following the rocedure of the reviou ection we get the exreion for dilaceent icrorotation teerature field force tree and coule tre by taking in euation (.)-(.). ( τ ) ( τ ) i j Particular Cae I : Neglecting icroolarity effect i.e. α β γ K j in euation (.)- (.) the exreion for dilaceent coonent force tree and teerature field are obtained in a theroelatic ediu a { e e e } u % ( ) a E a E a E u % ( ) b E e b E e b E e { } T% ( ) E e E e c E e ( ) e ( ) e ( ) e % t ( ) { a ι b E a ι b E a ι b E } 7

11 { ( ) } { ι 8 ( τ ) } { ι 8 ( τ ) } % t ( ) [ b ι a τ E e 8 ( a ) F a E E E E b a E e b a E e ] ( ) F a a E ( ) F a a E {( ) ( ) ( )} a b a b a b a b a b a b a [ { ( τ ) } ε ( τ Ξ )( τ ) }] b ι [{ ( τ ) }( ) ( )( ) } ] ε τ τ Ξ ( τ Ξ) {( ) } ε and ( ) ± are the root of the euation σ σ σ ε σ { ( τ ) } ( τ )( τ ) Ξ

12 σ { ( τ )} ε Ξ { ( τ ) } ( τ )( τ ) ε ε { ( τ ) } τ ( Ξ )( τ ) ( τ Ξ )( τ ) With λ µ ρc µ ρ C i. For L-S theory : Taking τ Ξ in exreion given by (.9)-(.) of articular cae I we obtain exreion for dilaceent coonent teerature field and force tree. ii. For G-L theory : Taking Ξ in exreion given by (.9)-(.)of articular cae I we obtain exreion for dilaceent coonent teerature field and force tree iii. For G-N theory : Neglecting icroolarity effect i.e. ( α β γ K j ) in ubcae III of cae I we get the exreion for dilaceent coonent teerature field and force tree are obtained in a theroelatic ediu by taking τ τ ε τ Ξ C ε ω h T ν ε ρ K in euation (.9)-(.) a { } e e e u % ( ) a E a E a E u % ( ) b E e b E e b E e { } T% ( ) E e E e c E e ( ) ( ) ( ) % t ( ) { a b E a b E a b E } 7 ι e ι e ι e { ( )} ( ) { b ι8a ( τ ) } E e ] { } % t ( ) [ b a E b a E ι 8 τ e ι 8 τ e

13 E ( a ) F a E E E ( ) F a a E ( ) F a a E {( ) ( ) ( )} a b a b a b a b a b a b { ( ) } a [ τ ε { ( )}( ) ε ι b [ } ε and ( ) {( ) } ε ε ± are the root of the euation σ σ σ σ ε ( ) σ ( ) ε ( ) ε ε ( )

14 where f % e and f % are even and odd art of the function f% ( ) reectively. Thu exreion (.) give u the Lalace tranfor f ( x ) of the function f (x t). Following Honig and Hirde (98) the Lalace tranfor function f ( x ) can be inverted to f (x t). σ ε ( ) Thu the exreion given by euation (.)-(.) with the hel of (.)-(.) and (.7) rereent the olution of lane train roble under conideration in the tranfored doain uing eigen value aroach.. Inverion of the tranfor To obtain the olution of the roble in the hyical doain we ut invert the tranfor for three theorie that i L-S G-L and G-N. Thee exreion are function of the araeter of Lalace and Fourier tranfor and reectively and hence are of the for f ( x ). To get the function f ( x t) in the hyical doain firt we invert the Fourier tranfor uing f ( x ) ex( i x) f ( ) d π { co( x) f iin( x) f } The lat te in the inverion roce i to evaluate the integral in euation (.). Thi wa done uing Roberg integration with adative te ie. Thi ethod ue the reult fro ucceive refineent of the extended traeoidal rule followed by extraolation of the reult to the liit when the te ie tend to ero. The detail can be found in Pre et al. (98)..Nuerical Reult and Dicuion Following Eringen [98] we take the following value of relevant araeter for the cae of Magneiu crytal a e d ρ.7 g µ. T C h c γ.779 c dyne dyne j. K. ε.7. τ -. c dyne c ec λ 9. C K τ dyne c. Call g C. 8.7 cal / c ec ec

15 > U (x) L-S(MTE) L-S(TE) G-L(MTE) G-L(TE) G-N(MTE) G-N(TE) -----> x Fig. Variation of noral dilaceent U (x) -----> T (x) - L-S(MTE) L-S(TE) G-L(MTE) G-L(TE) G-N(MTE) G-N(TE) ---> x - - Fig. Variation of noral force tre T (x)

16 L-S(MTE). G-L(MTE) G-N(MTE) -----> M (x). ----> x -. - Fig. Variation of tangential coule tre M (x) L-S(MTE) L-S(TE) G-L(MTE) G-L(TE) G-N(MTE) G-N(TE) -----> T * (x) > x Fig. Variation of terature field T * (x) Dicuion The variation of noral dilaceent U with ditance x for three different theorie (L-S G-L and G-N) in both edia after ultilying the original value for G-N theory in MTE ediu by

17 are hown in fig. The value of noral dilaceent due to icrorotation effect are le in MTE ediu in coarion to TE ediu in the x. for all three theorie wherea the value of U ocillate a x increae further in the ret of the range for both edia. It i alo evident that noral dilaceent decreae for both edia for L-S and G-L theorie increae gradually in MTE ediu for G-N theory and ocillate in TE ediu for G-N theory. The value of noral force tre T in agnitude are ore for three different theorie in MTE ediu in coarion to TE ediu. It i alo noticed that the value of noral force tre ocillate for L-S and G-L theorie in MTE and TE edia. The value of noral force tre alo ocillate for G-N theory in TE ediu wherea thee decreae gradually with increaing value of x in MTE ediu. Thee variation of noral force tre have been hown in fig. after dividing the original value by in cae of G-N theory in MTE ediu. Fig. deict the variation of tangential coule tre M for three different theorie in MTE ediu after dividing the original value for G-N theory by. The behaviour of tangential coule tre i ocillatory for three theorie. It i noticed that the value of tangential coule tre for G-N theory are large in coarion to L-S and G-L theorie in the range x. and the value are all for the ret of the range. The range of value of teerature field in agnitude i large in cae of three theorie in MTE ediu in coarion to TE ediu. It i alo oberved that teerature field ocillate in TE ediu for three different theorie but in MTE ediu for L-S and G-L theorie the teerature field ocillate. The value of teerature field for G-N theory decreae gradually with increaing value of x in MTE ediu. Thee variation hown in fig. after ultilying the original value in cae of L-S and G-N theorie by and reectively in MTE ediu; the original value in cae of G-N theory (TE ediu) and alo agnified by ultilying. Concluion Fro the above nuerical reult we conclude that icroolarity ha a ignificant effect on noral dilaceent noral force tre and teerature field echanicalource for three theorie.mcroolar effect i ore areciable for noral dilaceent and teerature field in coarion to noral force tre. Alication of the reent aer ay alo be found in the field of teel and oil indutrie. The reent Proble i alo ueful in the field of geoechanic where the interet i about the variou henoenon occurring in the earthuake and eauring of dilaceent tree and teerature field due to the reence of certain ource. Noenclature λ µ Lae' contant α β γ K Microolar aterial contant α λ λ tretch. Material contant due to the reence of

18 λ I µ I K I α I ν γ I α I λ I λ I Microtretch vicoelatic contant ρ Denity j Micro-inertia u Dilaceent vector Microrotation vector Scalar icrotretch t ij ij Force tre tenor Coule tre tenor λ l δ ij ε ijr ι Iota Microtre tenor Kronecker delta Alternating tenor Gradient oerator And dot denote the artial derivative w.r.t. tie. Reference : Bochi E. and Jean D.(99): A Generalied Theory of Linear MicroolarTheroelaticity- Meccanica Vol. VIII.-7. Bakhi R. Bera R.K. and Debnath L.(): Eigen value aroach to tudy the effect of rotation and relaxation tie in two dienional roble of generalied theroelatic Int. J. Engg. Sci. Vol Craciu L.(99): On the uniuene of the olution of oe boundary value roble of oe boundary value roble of the linear icroolartheroelaticity - Bull. Int. Politech. Iai. Sect.Vol Eringen A.C.(97):Foundation of MicroolarTheroelaticity- International Center for Mechanical Science Coure and Lecturer. No.. -Sringer Berlin. Eringen A.C.(98): Plane wave in a non-local icroolar elaticity -Int. J. Engg. Sci. Vol.. -. Eringen A.C.(98):Theory of icroolar elaticity in Fracture (chater-7) Vol.IIAcadeic Pre New York Ed. H. Leibowit. Green A.E. and Linday K.A. (97): Theroelaticity - J. Elaticity Vol.. -. Green A.E. and Naghdi P.M. (977): On therodynaic and the nature of the econd law Proc. Ray Soc. London A. Vol

19 Green A.E. and Naghdi P.M. (99): A Re-exaination of the Baic Potulate of Theroechanic- Proc. Ray. Soc London A Vol Green A.E. and Naghdi P.M.(99): Theroelatic without energy diiation J. Elaticity Vol Green A E and Linday K. A.(97):Theroelaticity -J. Elaticity Vol.. -. Honig G. and Hirde U. (98): A ethod for the nuerical inverion of the Lalace tranfor - J. Co. Al. Math Vol..-. Lord H. W. and Shulan Y.(97): A generalied dynaical theory of theroelaticity- J. Mech. Phy. Solid Vol Kuar R. and ChadhaT.K. (98): On torional loading in an axietricicroolar elatic ediu - Proc. Indian Acad. Sci.(Math Sci.) Vol Kuar R. and Dewal S. (): Mechanical and theral ource in the icroolargeneralied theroelatic ediu J. Sound and VibrationVol Kuar R. and Rani L. (): Deforation due to echanical and theral ource in the generalied orthorhobic theroelatic aterial-sadhana Vol Kuar R. and RaniL.(): Deforation due to echanical and theral ource in the generalied half-ace with void -Journal of Theral StreVol Mahalanabi and Manna (997): Eigen value aroach to the roble of linear icroolartheroelaticity- J. Indian Acad. Math. Vol MarinM. andluu M.(998): On haronic vibration in theroelaticity of icroolar bodie - J. Vibration and Control Vol Nowacki W. (9): Coule tree in the theory of theroelaticity - Proc. ITUAM yoia Sringer-Verlag Pre W. H. Teukolky S.A. Vellerling W.T. and Flannery B. P.(98): Nuerical Recie - Cabridge Univerity Pre Cabridge. Paarella F.(99):Soe reult in icroolartheroelaticity - Mechanic ReearchCounication Vol Sturnin D.V. ():On characteritic tie in generalied theroelaticity - J. Al. Math. Vol Shara J.N. and Chand D.(99): On the axiyetric and lane train roble of generalied theroelaticity- Int. J. Engg.Sci. Vol.-. -.