Reflection and Transmission of Plane Waves at Micropolar Piezothermoelastic Solids

Transcription

1 Journal of Sold Mechancs Vol. 9 No. (7 pp Reflecton and Transmsson of Plane Waves at Mcropolar Pezothermoelastc Solds R. Kumar M. Kaur Department of Mathematcs Kurukshetra Unversty Kurukshetra 69 Inda Department of Mathematcs Sr Guru Teg Bahadur Khalsa College Anandpur Sahb Punjab 48 Inda Receved 5 June 7; accepted August 7 ABSTRACT The present nvestgaton analyss a problem of reflecton and transmsson at an nterface of two mcropolar orthotropc pezothermoelastc meda. The basc equatons and consttutve relatons for mcropolar orthotropc pezothermoelastc meda for G-L theory are derved. The epressons for ampltude ratos correspondng to reflected and transmtted waves are derved analytcally. The effect of angle of ncdence frequency mcropolarty thermopezoelectrc nteractons on the reflected and transmtted waves are studed numercally for a specfc model. Some specal cases of nterest one are also deduced. 7 IAU Arak Branch. All rghts reserved. Keywords: Orthotropc; Mcropolar; Pezothermoelastc; Ampltude ratos; Angle of ncdence. INTRODUCTION M ICROPOLAR elastcty theory whch takes nto consderaton the granular character of the medum descrbes deformaton by a mcrorotaton and a mcrodsplacement. Erngen frst showed that the classcal elastcty theory [4] and the coupled stress theory [] are two specal cases of mcropolar elastcty. The lnear theory of mcropolar thermoelastcty was developed by etendng the theory of mcropolar contnua to nclude thermal effects by Nowack [9] and Erngen []. A comprehensve revew on the mcropolar theromoelastcty s gven by Erngen []. In most of the engneerng problems ncludng the response of sols geologcal materals and compostes some sgnfcant features of the contnuum response may not take nto account by the assumptons of sotropc behavor. The formulaton and soluton of ansotropc problems s far more dffcult and cumbersome than ther sotropc counterparts. Number of researchers pad attenton to the elastodynamc response of an ansotropc contnuum n the last few years. In partcular transversely sotropc and orthotropc materals whch may not be dstngushed from each other n plane stran and plane stress cases have been more regularly studed. The statc problems of plane mcropolar stran of a homogeneous and orthotropc elastc sold torson problems of homogeneous and orthotropc cylnders n the lnear theory of mcropolar elastcty and bendng of orthotropc mcropolar elastc beams by termnals couple were studed by Iesan [--]. Fnte element method for orthotropc mcropolar elastcty was developed by Nakamura et al. [8]. Kumar & Choudhary [-4-5] and [6-7] have studed varous problems n orthotropc mcropolar contnua. Pezoelectrc ceramcs and compostes fnd applcatons n many engneerng applcatons e.g. sensors actuators ntellgent structures rocket propelled grenades ultrasonc magng when thermal effects are not consdered. Correspondng author. E-mal address: mandeep5@yahoo.com (M.Kaur. 7 IAU Arak Branch. All rghts reserved.

2 R.Kumar and M.Kaur 59 Pezoelectrc ceramcs and pezoelectrc polymers are pyroelectrc meda whch are used n small structure and ntellgent system. The thermo-pezoelectrc materal response entals an nteracton of three major felds namely mechancal thermal and electrc n the macro-physcal world. The thermopezoelectrc materal has one mportant applcaton to detect the responses of a structure by measurment of the electrc charge sensng or to reduce ecessve responses by applyng addtonal electrc forces or thermal forces actuatng. Intellgent structure can be desgned by ntegratng sensng and actuatng. The thermopezoelectrc materals are also often used as resonators whose frequences need to be precsely controlled. It s mportant to quantfy the effect of heat dsspaton on the propagaton of wave at low and hgh frequences due to the couplng between the thermoelastc and pyroelectrc effects. The theory of thermo-pezoelectrcty was frst developed by Mndln []. The physcal laws for the thermo-pezoelectrc materals have been eplored by Nowack [--]. Chandrasekharaah [4-5] has generalzed Mndln s theory of thermo-pezoelectrcty to account for the fnte speed of propagaton of thermal dsturbances. Chen [4] derved the general soluton for transversely sotropc pezothermoelastc meda. Hou et al.[] constructed Green s functon for a pont heat source on the surface of a sem-nfnte transversely sotropc pyroelectrc meda. Abd-Alla et al. [8] nvestgated reflecton and refracton of plane quaslongtudnal waves at an nterface of two pezoelectrc meda under ntal stresses. Pang et al. [6] dscussed the reflecton and refracton of plane waves at the nterface between two transversely sotropc pezoelectrc and pezomagnetc meda. Dfferent researchers have studed the problems of reflecton n pezoelectrc meda notable among them are Sharma et al. [8] Abdalla and Alshakh [8-9] Kuang and Yuan[7] Abdalla et al. [] Alshakh [6-7] Abd-Alla Hamdan Gorgo and Vescovo [6] Guo and We [5] Othman [9] Othman Atwa Hasona and Ahmed [] Abd-Alla Gorgo Galantucc Hamdan and Vescovo [7]. In the present paper the reflecton and transmsson phenomenon of plane waves at an nterface of two orthotropc mcropolar pezothermoelastc meda has been dscussed. It s found that there est fve plane waves n mcropolar orthotropc pezothermoelastc medum namely quas longtudnal dsplacement wave quas thermal wave quas coupled transverse dsplacement and quas mcrorotatonal waves and one wave mode correspondng to electrc potental wave. When a plane quas wave s ncdent at an nterface the ampltude ratos of varous reflected and transmtted waves are computed numercally and are plotted graphcally wth angle of ncdence. BASIC EQUATIONS Followng Green-Lndsay [5] the basc equatons of homogeneous orthotropc mcropolar pezothermoelastc sold wth two relaaton tmes n the absence of body forces body couples electrc charge densty and heat sources are gven by (a Consttutve relatons t C A w g E ( T T ( kl jkl kl jkl kl jk k j m D w A e E ( j jkl kl jkl kl jk k D E g ( T T ( j j jk jk q T b T k e (4 j j The deformaton and wryness tensor are defned as followng: u w w w (5 j j jk k j j t (b Balance laws kl k u (6 m t Jw (7 kl k lmn mn 7 IAU Arak Branch

3 5 Reflecton and Transmsson of Plane Waves at Mcropolar. D (8 q T S (9 where S b e E c ( T T j j where tkl m kl are the stress tensor couple stress tensor; D s the electrc dsplacement vector E s the electrc feld vector q s the heat flu vector; S s the entropy; T s the thermodynamc temperature; T s the absolute temperature; c s the specfc heat at constant stran; v and are thermal relaaton tmes; s the bulk mass densty; J s the mcronerta; u l and w k are the components of dsplacement vector and mcrorotaton vector respectvely; j s delectrc modul; s pyroelectrc modul; k j s thermal conductvty tensor; b are the coeffcents characterzng the lack of a centre of symmetry j are the components of mcro-stran tensor jk s the permutaton tensor s the thermal elastc couplng tensor; C G D are the characterstc constants of materal; jk kl jkl jkl jkl g s the electro-elastc couplng modul where C D g satsfy the symmetrc relatons jkl jkl jk C C D D g g. ( jkl klj jkl klj jk kj In a centrosymmetrc bodes all components of A jkl vansh. FORMULATION OF THE PROBLEM By usng the transformatons followng Slaughter [] on the set of Eqs. ( to (9 the equatons for mcropolar orthotropc pezothermoelastc medum are derved. We consder an nterface of two homogeneous centrosymmetrc orthotropc mcropolar pezothermoelastc meda ntally n an undeformed state and at unform temperature T namely medum M and medum M. We take the orgn of coordnate system on the plane nterface and as pontng vertcally nto the medum M s taken whch s desgnated as. Plane waves are consdered such that all the partcles on a lne parallel to as are equally dsplaced so that all the partal dervatves wth respect to the varable wll be zero. Therefore we take u ( u u w ( w E ( E E E s the electrc potental and so that the feld equatons and consttutve relatons reduce to the followng: u u u w u C C ( C C ( C C g g ( T T t ( u u u w u C C ( C C ( C C g g ( T T t ( w w u u w D D ( C C ( C C ( C C C w ( g g J t ( u u u ( (. g 7 g g g 9 T T (4 7 IAU Arak Branch

4 R.Kumar and M.Kaur 5 T T u u k k T ( T ( c ( T T. t t t (5 u u. t C C g ( T T (6 u u t C C g (7 m D 86 w (8 where C C9 C9 C99 are the coeffcents of lnear thermal epanson. We have used the notatons for the materal constants. For convenent we ntroduce the followng dmensonless quanttes C c = = u = u u = u w w t t m m ' ' ' ' ' ' ' j j j j c c c c C C D4 T C C T D D t t c ' ' ' ' ' ' T cg g J (9 where s the characterstc frequency of the materal and c s the longtudnal wave velocty of the medum. By usng the dmensonless quanttes n Eqs. (-(5 we obtan the followng equatons u u u w u (. a a a a4 a5 T T a6 t ( u u u w u (. a7 a8 a 9 a a a T T a t ( w w u u w a4 a5 a 6 a7w a8 a9 t ( u u u (. a a a a a T T ( T T u u ( ( ( (. a4 a 5 a6 a7 a8 T T t t t (4 where 7 IAU Arak Branch

5 5 Reflecton and Transmsson of Plane Waves at Mcropolar. C C C ( C C C ( g g g T c a a a a a a C C ( C C C C C C C ( C C C g g g g T a a a a a a C 7 C 7 C 7C C 7 C 7 C 7 c a4 C 7 a D ( C C C c ( C C C c ( C C C c D C D C D D a5 a 6 a 7 4 ( 4 ( 4 4 ( g g C g c Jc g g g g a a a a a a CD4 D4 g g g a T k c c c g c c a4 a5 a 6 a 7 a 8 g k k k k k (5 4 PLANE WAVE PROPAGATION We consder harmonc plane wave propagatng n the plane at a gven frequency as: ( ( u u w T ( u u w T e t k (6 v k where u u w T are functons of only k s the wave number and c where c( v s the phase sn velocty of wave propagatng n plane along a drecton makng an angle wth as. Usng Eq. (6 n Eqs. (-(4 a system of fve homogeneous equatons s obtaned n fve unknowns u u w T whch for non-trval soluton yeld d d d d d A A A 8 A 6 4 A 4 5 A 6 d d d d d (7 where A a a a a a A a ( h a a h a a h a h a h a a a a a a h a A ( h a a h a a h a a h a ( k a a a a h h a h h k h a h h a h a h a h ( a h a h a h a a h a a h h A ( k a a a a h h a h k h h a h h a ( h a h h k h7 ah5 hk h 6 h h5 h4k ah5 ah ( a7h 6h h7 a h h a ( h a h h h h A ( k h h a h k h h a h h a ( h k h k h h6 h5k ah6 ah ( ah 8h h8 a4 8h 6h h4 ( 6a4 a7 A ( h k h k h h k ( h h h a k a h a a h ( a a a k ( a a h a a a h5 7a6 k h6 5a4 aa a 5 aa h7 7( a ( 5 7a5 a aa5 8 a a 8 h h ( a a a k a h ( a a ( a k k IAU Arak Branch

6 R.Kumar and M.Kaur 5 h a a h a a h h a a h a a k a h6 474a6 k h7 9 ( a6 5a 9 ( 4a 6 9 ( 5 4a 4 5( a5 6a ( a ( a h ( k k ( k ( h a a a h ( a a k h h a a a h ( a a a h k h a ( ka a ( a ( ka h ( ( ka ( ka h a ( a a a k a k a k a k ( a k a k a k a k a ( a a k a k a k a ( k a k a ( a k a a The roots of the Eq. (7 gve the veloctes of fve plane waves n the decreasng order of the veloctes.e quas longtudnal dsplacement wave (quas LD wave quas thermal wave (quas T wave quas CD-I quas CD-II wave and electrc potental wave (PE wave. 5 REFLECTION AND TRANSMISSION We consder an nterface of two homogeneous orthotropc mcropolar pezothermoelastc meda n contact wth each other. A plane wave quas LD wave or quas thermal wave s ncdent makng an angle wth as at an nterface. Each ncdent wave results n fve reflected wave modes n medum M whch s desgnated as and fve transmtted wave modes n medum M whch s desgnated as. In medum M and medum M reflected and transmtted wave modes are represented by quas LD wave quas thermal wave quas CD-I wave quas CD-II wave and one other mode correspondng to electrc potental wave mode.e. PE wave mode. All the quanttes n medum M are denoted by bar. The values of dsplacements mcrorotaton electrc potental and temperature dstrbuton n medum M are gven by u ( ( B e B e B e B e B e B e B e B e B e B e e 5 ( t k (8 u ( ( m B e m B e m B e m B e m B e m B e m B e m B e m B e m B e e 4 5 ( t k (9 w ( ( n B e n B e n B e n B e n B e n B e n B e n B e n B e n B e e 4 5 ( t k ( ( ( g B e g B e g B e g B e g B e g B e g B e g B e g B e g B e e 4 5 ( t k ( T ( ( l B e l B e l B e l B e l B e l B e l B e l B e l B e l B e e 4 5 ( t k ( 7 IAU Arak Branch

7 54 Reflecton and Transmsson of Plane Waves at Mcropolar. and for medum M : u ( ( B e B e B e B e B e e ( 4 5 ( t k 4 5 u ( ( m B e m B e m B e m B e m B e e (4 4 5 ( t k w ( ( n B e n B e n B e n B e n B e e (5 4 5 ( t k ( ( g B e g B e g B e g B e g B e e (6 4 5 ( t k T ( ( l B e l B e l B e l B e l B e e (7 4 5 ( t k where 4 5 are the veloctes of reflected quas LD wave quas T wave quas CD-I wave quas CD-II wave and PE wave mode respectvely n medum M and 4 5 are the veloctes of transmtted quas LD wave quas T wave quas CD-I quas CD-II wave and PE wave mode respectvely n medum M. and 4 m n g l 45 a a a6k a4 ( 5 6 a4 7 a a k a a a5 a4 a k a( 4 6 a4 7 a a a a k 5 6 a a k a( a4 7 a a5 a6k a4 a a( 4 5 a4 7 a a a5 a6k a4 4 a a k (8 7 IAU Arak Branch

8 R.Kumar and M.Kaur 55 6 BOUNDARY CONDITIONS The approprate boundary condtons at an nterface are gven by T T t = t t = t m = m k k u = u u = u w w T T E E D D (9 Makng use of Eqs. (8 to (7 n boundary condtons gven by Eq. (9 we obtan a system of ten homogeneous equatons as: 5 aj B j = ; =... (4 j = where a d k d m d g d ( l a d k d m d g d ( l 4 j 4 a d k ( d m d g d ( l k 4 a a ( d g d m k a a ( d g d m k a a ( d m d g k 6 5 j 6 5 k 5 6 a d 7 n a j d 7 n ak d 7 n a4 l a4 j l a4k d 8 l a a a a m a a m 5 5 j 5k 6 6 j 6k a n a a n a l a a l 7 7 j 7k 8 8 j 8k a kg a a k g a d g kd d m a ( 9 9 j 9k 9 a d g j 9 kd d m a ( a d g k d d m a ( k j k 45 (4 where C C g g T C g g d d d d d d C C C C C C D C k g g d d d d d D4C k g g C C g g T C g g d d d d d d C C C C C C D C g g d d d d D4C g g when quas LD wave s ncdent: B B B4 B5. Dvdng the set of equatons throughout by B we obtan a system of ten non-homogeneous equatons n ten unknowns whch can be solved by Crammer s rule and we have Z B 5 B ; when quas T wave s ncdent: B B B4 B5. Dvdng the set of equatons throughout by B we obtan a 7 IAU Arak Branch

9 56 Reflecton and Transmsson of Plane Waves at Mcropolar. system of ten non-homogeneous equatons n ten unknowns whch can be solved by Crammer s rule and we have Z B 5 B ; p where a and (...( p... can be obtaned by replacng respectvely 5 the st nd th T columns of by [ a p a p a p a4 p... a p ]. 7 PARTICULAR CASES If we neglect pezoelectrc effect n medum M we obtan ampltude ratos at an nterface of orthotropc mcropolar pezothermoelastc sold and orthotropc mcropolar thermoelastc sold wth the values of aj as: a d k d m d g d ( l a d k d m d g d ( l 4 j 4 a d k ( d m d g d ( l k 4 a a ( d g d m k a a ( d g d m k a a ( d m d g k 6 5 j 6 5 k 5 6 a d 7 n a j d 7 n ak d 7 n a4 l a4 j l a4k d 8 l a a a a m a a m 5 5 j 5k 6 6 j 6k a n a a n a l a a l 7 7 j 7k 8 8 j 8k a d g kd d m a ( a d g kd d m a ( a j 9 9k 4 5 j k 4 By neglectng the mcropolarty effect n medum M we obtan ampltude ratos at an nterface of orthotropc mcropolar pezothermoelastc sold and orthotropc pezothermoelastc sold wth the values of aj as: a d k d m d g d ( l a d k d m d g d ( l 4 j 4 a d k ( d m d g d ( l k 4 a a ( d g d m k a a ( d g d m k a a ( d m d g k 6 5 j 6 5 k 5 6 a d 7 n a j d 7 n ak a4 l a4 j l a4k d 8 l a a a a m a a m 5 5 j 5k 6 6 j 6k a l a a l a kg a a k g 7 7 j 7k 8 8 j 8k a d g kd d m a ( a d g kd d m a ( a j 9 9k d g k d d m a ( j k 4 If we neglect pezoelectrc and mcropolarty effects n medum M and M our results tally wth those obtaned for perfect bondng case. 7 IAU Arak Branch

10 R.Kumar and M.Kaur 57 8 NUMERICAL RESULTS AND DISCUSSION In order to determne the ampltude ratos the method of crammer s rule has been used and computer program n Matlab 7.8 has been developed. The physcal data for medum M s gven by C 7.46 Nm C.9 Nm C.7 Nm C 8.9 Nm C.99 Nm C.8 Nm C.4 Nm C. Nm g.4 cm g.65 cm g.5 cm Nm / K 9.7 Nm / K 7.6 cm / K.8 s 6.9 cm k 9.5 Wm K k 9.7 Wm K.67 C N m.58 C N m s T K Kgm c NmK D.4 N D.4 N 4 86 g g K J. m and for medum M s gven by C.4 Nm C.786 Nm C 6. Nm C.8 Nm C 75.8 Nm C 6.9 Nm C 6.6 Nm C 6. Nm g. cm g.6 cm g.57 cm g cm.5 Nm / K.78 Nm / K 8. cm / K.6 s k. Wm K k. Wm K.56 C N m.55 C N m s T K 75 Kgm c 5 NmKg K D.9 N D.7 N 4 86 J. m Fgs. - show the varatons of ampltude ratos wth angle of ncdence for ncdence of plane waves at an nterface for GL-theory. In Fgs. - corresponds to ampltude ratos n orthotropc mcropolar pezothermoelastc sold corresponds to ampltude ratos n orthotropc mcropolar thermoelastc sold corresponds to ampltude ratos n orthotropc pezothermoelastc sold. 8. Quas LD wave ncdence Fgs. - represent the varatons of ampltude ratos Z ; for ncdence of quas LD wave. Fg. shows that the values of ampltude rato Z for and ncrease from normal ncdence to grazng ncdence. It s notced that the values of ampltude rato for are less than the values for. It shows that the mcropolarty effect ncreases the magntude of ampltude rato. The values of ampltude rato Z for are magnfed by multplyng by. It s notced clearly from Fg. that the values of ampltude rato Z for start wth mnmum value at normal ncdence and then ncrease gradually to attan mamum value at the grazng ncdence whle the values for decrease wth ncrease n angle of ncdence. The values of ampltude rato for orthotropc pezothermoelastc sold are greater than the values for orthotropc mcropolar thermoelastc sold that reveals the mcropolarty effect. From Fg. t s clearly revealed that the values of ampltude rato Z for ncrease n the whole range. It s observed that the values of ampltude rato for orthotropc mcropolar pezothermoelastc sold are greater than the values for orthotropc pezothermoelastc sold and orthotropc mcropolar thermoelastc sold. 7 IAU Arak Branch

11 Ampltude rato Ampltude rato 58 Reflecton and Transmsson of Plane Waves at Mcropolar. Fg. 4 depcts the varaton of ampltude rato Z 4 wth angle of ncdence. The values of ampltude rato for and ncrease whle oscllate wth ncrease n angle of ncdence. The values of ampltude rato for are greater than the values of ampltude rato for. Fg. 5 shows that the values of ampltude rato Z 5 for and attan mnmum value at normal ncdence and then the values ncrease sharply as ncreases. The values for are greater than the values for and n the whole range. Fg. 6 shows that the values of ampltude rato Z 6 for ncrease wth angle of ncdence whle the values for decrease. In ths case the removal of mcropolarty effect ncreases the magntude of ampltude rato. Fg. 7 depcts that the values of ampltude rato Z 7 for and ncrease as angle of ncdence ncreases whle the values for decreases. It s notced from Fg. 8 that the values of ampltude rato Z 8 for and ncrease wth ncrease n angle of ncdence. The values for are greater than the values for n the whole range. Fg. 9 reveals that the values of ampltude rato Z 9 for attan mamum value at normal ncdence and then decrease n the further range and are greater than the values for n the whole range. It s depcted from Fg. that the values of ampltude rato Z for ncrease n the whole range. The values for are less than the values for n the whole range Angle of ncdence Fg. Varaton of ampltude rato Z wth angle of ncdence Angle of ncdence Fg. Varaton of ampltude rato Z wth angle of ncdence. 7 IAU Arak Branch

12 Ampltude rato Ampltude rato Ampltude rato Ampltude rato R.Kumar and M.Kaur Angle of ncdence Fg. Varaton of ampltude rato Z wth angle of ncdence Angle of ncdence Fg.4 Varaton of ampltude rato Z 4 wth angle of ncdence Angle of ncdence Fg.5 Varaton of ampltude rato Z 5 wth angle of ncdence Angle of ncdence Fg.6 Varaton of ampltude rato Z 6 wth angle of ncdence. 7 IAU Arak Branch

13 Ampltude rato Ampltude rato Ampltude rato Ampltude rato 5 Reflecton and Transmsson of Plane Waves at Mcropolar Angle of ncdence Fg.7 Varaton of ampltude rato Z 7 wth angle of ncdence Angle of ncdence Fg.8 Varaton of ampltude rato Z 8 wth angle of ncdence Angle of ncdence Fg.9 Varaton of ampltude rato Z 9 wth angle of ncdence Angle of ncdence Fg. Varaton of ampltude rato Z wth angle of ncdence. 7 IAU Arak Branch

14 Ampltude rato R.Kumar and M.Kaur 5 8. Quas T wave ncdence Fgs. - represent the varatons of ampltude ratos Z ; wth angle of ncdence for ncdence of quas T wave. Fg. shows that the values of ampltude rato Z for start wth mamum value at normal ncdence and then the values decrease to attan mnmum value at the grazng ncdence. It s seen that the values for ncrease wth angle of ncdence. Fg. depcts the varaton of ampltude rato Z wth angle of ncdence. The values of ampltude rato for decrease wth angle of ncdence to attan mnmum value at the grazng ncdence. In ths case the removal of pezoelectrc effect and the removal of mcropolarty effect ncreases the magntude of ampltude rato. It s notced from Fg. that the values of ampltude rato Z for get ncreased whle the values for get decreased wth angle of ncdence. It s seen that absence of mcropolarty effect rases the magntude of ampltude. Fg. 4 shows that the values of ampltude rato Z 4 for reman less than the values for and n the whole range. Fg. 5 depcts that the values of ampltude rato Z 5 for decrease n the whole range ecept the ntal range where the values of ampltude rato get ncreased. The values of ampltude ratos for are greater than the values for n the whole range. Fg. 6 shows that the values of ampltude rato Z 6 for and are very small n magntude and oscllate from normal ncdence to grazng ncdence. The absence of mcropolarty effect ncreases the magntude of ampltude rato n ths case. It s seen from Fg. 7 that the values of ampltude rato Z 7 for get ncreased wth ncrease n. The values for ncrease n the whole range ecept the ntal range where the values decrease. It s depcted from Fg. 8 that the values of ampltude rato Z 8 for start wth mnmum value at the normal ncdence and then ncrease to attan mamum value near the grazng ncdence. The values for decrease n the whole range. The values of ampltude rato n the absence of pezoelectrc effect are smaller than the values n the presence of pezoelectrc effect. Fg. 9 shows that the values of ampltude rato Z 9 for and attan mamum value at the normal ncdence and then decrease to attan mnmum value at grazng ncdence. Fg. depcts that the values of ampltude rato Z for ncrease sharply n the ntal range and then decrease as long as ncreases. In ths case the mcropolarty effect ncreases the magntude of ampltude rato n the whole range Angle of ncdence Fg. Varaton of ampltude rato Z wth angle of ncdence. 7 IAU Arak Branch

15 Ampltude rato Ampltude rato Ampltude rato Ampltude rato 5 Reflecton and Transmsson of Plane Waves at Mcropolar Angle of ncdence Fg. Varaton of ampltude rato Z wth angle of ncdence Angle of ncdence Fg. Varaton of ampltude rato Z wth angle of ncdence Angle of ncdence Fg.4 Varaton of ampltude rato Z 4 wth angle of ncdence Angle of ncdence Fg.5 Varaton of ampltude rato Z 5 wth angle of ncdence. 7 IAU Arak Branch

16 Ampltude rato Ampltude rato Ampltude rato Ampltude rato R.Kumar and M.Kaur Angle of ncdence Fg.6 Varaton of ampltude rato Z 6 wth angle of ncdence Angle of ncdence Fg.7 Varaton of ampltude rato Z 7 wth angle of ncdence Angle of ncdence Fg.8 Varaton of ampltude rato Z 8 wth angle of ncdence Angle of ncdence Fg.9 Varaton of ampltude rato Z 9 wth angle of ncdence. 7 IAU Arak Branch

17 Dsplacement(u Dsplacement(u Ampltude rato 54 Reflecton and Transmsson of Plane Waves at Mcropolar Angle of ncdence Fg. Varaton of ampltude rato Z wth angle of ncdence. Fgs. -4 show the varatons of wavefronts of dsplacement mcrorotaton and temperature wth respect to -as for tme nstants t and t 5 secs n orthotropc mcropolar pezothermoelastc half-space. Fgs. and depct that the ampltude of horzontal dsplacement u and mcrorotaton w decreases wth ncrease n the value of. The values for t 5 are greater than the values for t whch shows that as the tme ncreases the ampltude ncreases. Fgs. and 4 show the varaton of vertcal dsplacement u and temperature T wth respect to -as. It s observed that the values of vertcal dsplacement u and temperature T frst decrease and then ncrease AMP(t= AMP(t= Fg. Varaton of horzontal dsplacement u wth -as for ncdence of quas LD wave..4. AMP(t= AMP(t= Fg. Varaton of vertcal dsplacement u wth -as for ncdence of quas LD wave. 7 IAU Arak Branch

18 Temperature(T Mcrorotaton(w R.Kumar and M.Kaur 55 AMP(t= AMP(t= Fg. Varaton of mcrorotaton w wth -as for ncdence of quas LD wave..5. AMP(t= AMP(t= Fg.4 Varaton of temperature T wth -as for ncdence of quas LD wave. 9 CONCLUSIONS The reflecton and transmsson coeffcents of varous plane quas waves on ncdence of quas LD wave and quas T wave at an nterface of two orthotropc mcropolar pezothermoelastc meda are obtaned n the present paper. It s notced that reflecton and transmsson coeffcents are nfluenced by pezoelectrc and mcropolarty effect. It s seen that when quas LD wave s ncdent the values of ampltude ratos of reflected quas T wave n the absence of mcropolarty effect are greater that reveals the effect of mcropolarty. When quas T wave s ncdent the pezoelectrc effect ncreases the magntude of ampltude rato of reflected quas CD-II wave and transmtted quas CD-II wave modes. The problem nvestgated n ths paper has wde applcatons n sgnal processng and wreless communcaton n addton to mprovement of SAW wave devces and defence equpment. REFERENCES [] Erngen A.C.996 Lnear theory of mcropolar elastcty Journal of Mathematcs and Mechancs 5: [] Erngen A.C. 97 Foundatons of Mcropolar Thermoelastcty Course Held at the Department for Mechancs of Deformable Bodes Sprnger. [] Erngen A.C. 99 Mcrocontnuum Feld Theory I Foundatons and Solds Sprnger New York. [4] Erngen A.C. Suhub E.S. 964 Non-lnear theory of ample mcro-elastc solds Internatonal Journal of Engneerng Scence :89-. [5] Green A.E. Lndsay K.A. 97 Thermoelastcty Journal of Elastcty : -7. [6] Abd-Alla A.N. Hamdan A.M. Gorgo I. Del Vescovo D. 4 The mathematcal model of reflecton and refracton of longtudnal waves n thermo-pezoelectrc materals Archve of Appled Mechancs 84(9: [7] Abd-Alla A.N. Gorgo I. Galantucc L. Hamdan A.M. Del Vescovo D. 6 Wave reflecton at a free nterface n an ansotropc pyroelectrc medum wth nonclasscal thermoelastcty Contnuum Mechancs and Thermodynamcs 8(-: IAU Arak Branch

19 56 Reflecton and Transmsson of Plane Waves at Mcropolar. [8] Abd-Alla A.N. Alshakh F.A. 9 Reflecton and refracton of plane quas-longtudnal waves at an nterface of two pezoelectrc meda under ntal stresses Archve of Appled Mechancs 79(9: [9] Abd-Alla A.N. Alshakh F.A. 9 The effect of the ntal stresses on the reflecton and transmsson of plane quasvertcal transverse waves n pezoelectrc materals Proceedngs of World Academy of Scence Engneerng and Technology 8: [] Abd-Alla A.N. Alshakh F.A. Al-Hossan A.Y. The reflecton phenomena of quas-vertcal transverse waves n pezoelectrc medum under ntal stresses Meccanca 47(: [] Iesan D. 97 The plane mcropolar stran of orthotropc elastc solds Archwum Mechank Stosowanej 5: [] Iesan D. 974 Torson of ansotropc mcropolar elastc cylnders Zetschrft für Angewandte Mathematk und Mechank 54: [] Iesan D. 974 Bendng of orthotropc mcropolar elastc beams by termnal couples Analele Stntfce Ale Unverstat Ias : [4] Chandrasekharaah D.S. 984 A temperature-rate-dependent theory of thermopezoelectrcty Journal of Thermal Stresses 7: 9-6. [5] Chandrasekharaah D.S.988 A generalzed lnear thermoelastcty theory for pezoelectrc meda Acta Mechanca 7: [6] Alshakh F.A. The mathematcal modellng for studyng the nfluence of the ntal stresses and relaaton tmes on reflecton and refracton waves n pezothermoelastc half-space Appled Mathematcs (8: [7] Alshakh F.A. Reflecton of quas vertcal transverse waves n the thermo-pezoelectrc materal under ntal stress (Green- Lndsay Model Internatonal Journal of Pure and Appled Scences and Technology : 7-9. [8] Sharma J.N. Wala V. Gupta S.K. 8 Reflecton of pezo-thermoelastc waves from the charge and stress free boundary of a transversely sotropc half space Internatonal Journal of Engneerng Scence 46(:-46. [9] Othman M.I.A. 5The effect of rotaton on pezothermoelastc medum usng dfferent theores Structural Engneerng and Mechancs 56(4: [] Othman M.I.A. Atwa S.Y. Hasona W.M. Ahmed E.A.A. 5 Propagaton of plane waves n generalzed pezothermoelastc medum: Comparson of dfferent theores Internatonal Journal of Innovatve Research n Scence Engneerng and Technology 4(4: 9-. [] Hou P.F. Luo W. Leung Y.T. 8 A pont heat source on the surface of a sem-nfnte transverse sotropc pezothermoelastc materal SME Journal of Appled Mechancs 75:-8. [] Mndln R.D. 96 On the Equatons of Moton of Pezoelectrc Crystals Problems of Contnuum Mechancs SIAM Phladelpha. [] Kumar R. Choudhary S. Mechancal sources n orthotropc mcropolar contnua Proceedngs of the Indan Academy of Scences (Earth and Planetary Scences : -4. [4] Kumar R. Choudhary S. Influence of Green s functon for orthotropc mcropolar contnua Archve of Mechancs 54: [5] Kumar R. Choudhary S. Dynamcal behavor of orthotropc mcropolar elastc medum Journal of Vbraton and Control 8: [6] Kumar R. Choudhary S. Response of orthotropc mcroploar elastc medum due to varous sources Meccanca 8: [7] Kumar R. Choudhary S. 4 Response of orthotropc mcropolar elastc medum due to tme harmonc sources Sadhana 9: 8-9. [8] Nakamura S. Benedct R. Lakes R. 984 Fnte element method for orthotropc mcropolar elastcty Internatonal Journal of Engneerng Scence : 9-. [9] Nowack W. 966 Couple stress n the theory of thermoelastcty Irreversble Aspects of Contnuum Mechancs and Transfer of Physcal Characterstcs n Movng Fluds SprngerVerlag. [] Nowack W.978 Some general theorems of thermo-pezoelectrcty Journal of Thermal Stresses : 7-8. [] Nowack W. 979 Foundatons of Lnear Pezoelectrcty Electromagnetc Interactons n Elastc Solds Sprnger Wen. [] Nowack W. 98 Mathematcal Models of Phenomenologcal Pezo-Electrcty New Problems n Mechancs of Contnua Unversty of Waterloo Press Waterloo Ontaro. [] Slaughter W.S. The Lnearzed Theory of Elastcty Brkhauser Basel. [4] Chen W.Q. On the general soluton for pezothermoelastc for transverse sotropy wth applcaton ASME Journal of Appled Mechancs 67: [5] Guo X. We P. 4 Effects of ntal stress on the reflecton and transmsson waves at the nterface between two pezoelectrc half spaces Internatonal Journal of Solds and Structures 5(: [6] Pang Y. Wang Y. S. Lu J.X. Fang D. N. 8 Reflecton and refracton of plane waves at the nterface between pezoelectrc and pezomagnetc meda Internatonal Journal of Engneerng Scence 46: 98-. [7] Kuang Z.B. Yuan X.G. Reflecton and transmsson of waves n pyroelectrc and pezoelectrc materals Journal of Sound and Vbraton (6:-. 7 IAU Arak Branch