Reflection of Plane Waves from a Free Surface of an Initially Stressed Transversely Isotropic Dissipative Medium

Transcription

1 ppled Mathematcs 0 9- do:0.6/am Publshed Ole September 0 ( Reflecto of Plae Waves from a Free Surface of a Itally Stressed Trasversely Isotropc Dsspatve Medum bstract Baljeet Sgh Jyot rora Departmet of Mathematcs Post Graduate Govermet College Chadgarh Ida B.S..I.T.M lampur Fardabad Ida E-mal: bsghgc@gmal.com Receved Jue 0; revsed July 5 0; accepted July 0 The goverg equatos of a trasversely sotropc dsspatve medum are solved aalytcally to obta the speeds of plae waves. The approprate solutos satsfy the requred boudary codtos at the stress-free surface to obta the expressos of the reflecto coeffcets of reflected quas-p (qp) ad quas-sv (qsv) waves closed form for the cdece of qp ad qsv waves. partcular model s chose for umercal computato of these reflecto coeffcets for a certa rage of the agle of cdece. The umercal values of these reflecto coeffcets are show graphcally agast the agle of cdece for dfferet values of tal stress parameter. The mpact of tal stress parameter o the reflecto coeffcets s observed sgfcatly. Keywords: Trasversely Isotropc Dsspatve Medum Ital Stress Plae Waves Reflecto Reflecto Coeffcets. Itroducto We ca ot kow the Earth completely by assumg mere a elastc body. If we cosder varous addtoal parameters e.g. porosty tal stress vscosty dsspato temperature vods dffuso etc. the we ca uderstad better the teror of the Earth. Ital stresses a medum are caused by varous reasos such as creep gravty exteral forces dfferece temperatures etc. The reflecto of plae waves at free surface terface ad layers s mportat estmatg the correct arrval tmes of plae waves from the source. Varous researchers studed the reflecto ad trasmsso problems at free surface terfaces ad layered meda [-]. The study of reflecto of plae waves the presece of tal stresses as well as dsspato s terestg. Wth the help of Bot [] theory of cremetal deformato Selm [] studed the reflecto of plae waves at a free surface of a tally stressed dsspatve medum. I the preset paper we studed the problem o reflecto of plae waves at a stress-free surface of a tally stressed trasversely sotropc sold half-space wth dsspato. The reflecto coeffcets of reflected waves are computed umercally to observe the effect of tal stress.. Formulato of the Problem ad Soluto Followg Bot [] the basc dyamcal equatos of moto x-z plae for a fte tally stressed medum the absece of exteral body forces are s s u P x z z t () s s w P x z x t w u where s the desty x z s rotatoal compoet s j ( j = ) are cremetal stress compoets u ad w arethe dsplacemet compoets. Followg Bot [] the stress-strarelatos are u w s C P C P x z () u w w u s s C s C C z x z x where C j are the cremetal elastc coeffcets. Copyrght 0 ScRes. M

2 0 B. SINGH ET L. For dsspatve medum elastc coeffcets are replaced by the complex costats: C C C C C C C C C C C C R I R where C C C C C C C I C are real. Followg Fug [5] the stress ad stra compoets dsspatve medum are s s e u u e t t j j where ( j = ) ad beg the agular frequecy. Wth the help of Equatos () ad () the Equato () becomes R I u R I w s C C P C C P x z R I u w s s C C z x (5) R I w R I u s C C C C z x Wth the help of Equato (5) the Equato () becomes R u R R P w R P u C P C C C x z x z I u I I w I u uc C C C 0 x xz z R w R R P u C C C z x z R P w C w x I I u I w I w C C C C 0 xz x z The dsplacemet vector gve by () () (6) U d e (8) where () assgs a arbtrary drecto of propagato of waves d dd s the ut dsplacemet vector k ad c t X. s the phase factor whch s the ut propagato vector c s the velocty of propagato X = (x z) ad k s correspodg wave umber whch s related to the agular frequecy by k c. The dsplacemet compoets u ad w are wrtte as U u O w () (7) s u d w d Makg use of Equato (9) to the Equatos (6) ad (7) we obta a system of two homogeeous equatos whch as o-trval soluto f where k x z ct e (9) c c IP 0 (0) I D D P D D D R R P D C R C R R C C P D C C C C D R R I I R P R I I C C C C I I P I I P The roots c c correspod to quas-p (qp) waves ad quas-sv (qsv) waves respectvely. The above two roots gve the square of veloctes of propagato as well as dampg. Real parts of the rght had sdes correspod to phase veloctes ad the respectve mgary parts correspod to dampg veloctes of qp ad qsv waves respectvely. It s observed that both c ad c deped o tal stresses dampg ad drecto of propagato. I the absece of tal stresses ad dampg the above aalyss correspods to the case of trasversely sotropc elastc sold.. Reflecto of Plae Waves from Free Surface We cosder a tally stressed dsspatve half-space occupyg the rego z > 0 (Fgure ). I ths secto we shall drve the closed form expressos for the reflecto coeffcets for cdet qp or qsv waves. The dsplacemet compoets of cdet ad reflected waves are as j j () j j j j u xzt d e w xzt d e where j j Copyrght 0 ScRes. M

3 B. SINGH ET L. s Qk d cos e d s e e s k Qd cose Qd se e (5) I the x-z plae the dsplacemet ad stress compoets due to the reflected qsv wave ( s e cose ) are wrtte as u d e w d e Fgure. Geometry of the problem. k ct sexcos ez k ct sexcos ez k ct sexcos ez () k ct sexcos ez Here subscrpts ad correspod to cdet qp wave cdet qsv wave reflected qp wave ad reflected qsv wave respectvely. I the x-z plae the dsplacemet ad stress compoets due to the cdet qp wave ( s e cos e ) are wrtte as u d e w d e s Qk d cos e d s e e s k Q d cose Q d s e e () where Q C C Q C C R I R R Q C C Q C C. I the x-z plae the dsplacemet ad stress compoets due to the cdet qsv wave ( s e cose) are wrtte as u d e w d e s Qk [d cos e d s e ] e s k Q d cose Q d s e e () I the x-z plae the dsplacemet ad stress compo- due to the reflected qp wave ets ( s e cose) are wrtte as u d e w d e s Qk d cos e d s e e s k Q d cose Q d se e (6) The boudary codtos requred to be satsfed at the free surface z = 0 fx s e P 0 (7) fz s 0 The above boudary codtos are wrtte as s s s s P(e e e e ) 0 (8) s s s s 0 The Equatos () to (6) wll satsfy the boudary codtos (8) f the followg Sell s law holds s e s e s e s e (9) c c c c ad the followg relatos hold = 0 (0) where P ad L Q = 0 () 7 8 kl d cose d se kld cose d se kld cose d se kld se d cose k Q d cose Q d se 5 k Q d cose Q d se 6 k Q d cose Q d se 7 k Q d se Q d 8 cos e () Copyrght 0 ScRes. M

4 B. SINGH ET L. ) For cdet qp wave ( = 0) () ) For cdet qsv waves ( = 0) () For sotropc case C = + + P C = P = S the the above theoretcal dervatos reduce to Selm [].. Numercal Example For umercal purpose a partcular example of the materal s chose wth the followg physcal costats R 0 R 0 C.680 N m C.560 N m R 0 R 0 C N m C N m I 0 I 0 C.050 N m C N m I 0 I 0 C 0.50 N m C 0.50 N m 7.0 kg m. From Equatos () ad () the reflecto coeff- cdet qp ad qsv waves. The umercal values of cets of reflected qp ad qsv waves are computed for the the reflecto coeffcets of reflected qp ad qsv waves are show graphcally Fgure for cdet qp wave ad Fgure for cdet qsv wave. For P = 0 the reflecto coeffcet of reflected qp wave oscllates for the whole rage of the agle of c- P dece of qp wave as show by sold le Fgure. The varatos of reflecto coeffcets of qp wave for = ad P = also chage at each agle of cdece as show by sold le wth asters ad sold les wth tragles respectvely. The comparso of the varatos of reflecto coeffcets for P = 0 P = ad P = shows the sgfcat effect of tal stress o reflected qp wa ve for cdet qp wave. Smlarly reflected qsv s also affected sgfcatly due to the presece of tal stress as show Fgure. For cdet qsv wave the reflecto coeffcet of reflected qp wave frst creases to ts maxmum value ad the decreases to ts mmum value at agle e = 5 whe P = 0. Thereafter t oscllates as show by sold le Fgure. The varatos of reflecto coeffcets of qp wave for P = ad P = are smlar to that for P = 0. The comparso of sold le sold le wth asters ad sold le wth tragles shows the sgfcat effect of tal stress o reflected qp wave for cdet qsv wave. Smlarly reflected qsv s also affected sgfcatly due to the presece of tal stress as show Fgure. Fgure. Varato of the reflecto coeffcets of qp a d qsv waves agast the agle of cdece for cdece of qp wave. 5. Coclusos The reflecto from the stress-free surface of a trasversely sotropc dsspatve medum s cosdered. The expressos for the reflecto coeffcets of reflected qp ad qsv waves are obtaed closed form for the cdece of qp ad qsv waves. For a partcular materal these coeffcets are computed ad depcted graphcally agast the agle of cdece for dfferet values of tal stress parameter. From the fgures t observed that ) the tal stresses affect sgfcatly the reflecto coeffcets of all reflected waves. ) For cdet qp wave the crtcal agle for reflected qsv wave s observed at e = 5 ad for cdet qsv wave the crtcal agle for reflected qp wave s observed also at e = 5. ) The effect of tal stresses o the reflecto coeffcets s Copyrght 0 ScRes. M

5 B. SINGH ET L. Fgure. Varato of the reflecto coeffcets of qp ad qsv waves agast the agle of cdece for cdece of qsv wave. mmum at e = 5 for cdece qp wave ad at e = 5 for cdece qsv wave. 6. Refereces [] S. B. Sha Trasmsso of Elastc Waves through a Homogeous Layer Sadwched Homogeous Meda Joural of Physcs of the Earth Vol. No. 999 pp. -. do:0.9/jpe95.. [] R. N. Gupta Reflecto of Plae Waves from a Lear Trasto Layer Lqud Meda Geophyscs Vol. 0 No. 965 pp. -. do:0.90/.958 [] R. D. Tooly T. W. Specer ad H. F. Sagoc Reflecto ad Trasmsso of Plae Compressoal Waves Geophyscs Vol. 0 No. 965 pp [] R. N. Gupta Reflecto of Elastc Waves from a Lear Trasto Layer Bullet of the Sesmologcal Socety of merca Vol No. pp do:0.90/.96 [5] R. N. Gupta Propagato of SH-Waves Ihomogeeous Meda Joural of the coustcal Socety of merca Vol. No pp do:0./.9077 [6] H. K. charya Reflecto from the Free Surface of Ihomogeeous Meda Bullet of the Sesmologcal Socety of merca Vol. 60 No. 970 pp [7] V. Cervey Reflecto ad Trasmsso Coeffcets for Trasto Layers Studa Geophysca et Geodaetca Vol. 8 No. 97 pp do:0.007/bf06709 [8] B. M. Sgh S. J. Sgh ad S. D. Chopra Reflecto ad Refracto of SH-Waves ad the Plae Boudary betwee Two Laterally ad Vertcally Heterogeeous Solds cta Geophysca Vol pp [9] B. Sgh Effect of Hydrostatc Ital Stresses o Waves a Thermoelastc Sold Half-Space ppled Mathematcs ad Computato Vol. 98 No. 008 pp [0] M. D. Sharma Effect of Ital Stress o Reflecto at the Free Surfaces of sotropc Elastc Medum Joural of Earth System Scece Vol. 6 No pp do:0.007/s [] S. Dey ad D. Dutta Propagato ad tteuato of Sesmc Body Waves Itally Stressed Dsspatve Medum cta Geophysca Vol. XLV 998 pp [] M. M. Selm ad M. K. hmed Propagato ad tte- Nuato of Sesmc Body Waves Dsspatve Medum uder Ital ad Couple Stresses ppled Mathematcs ad Computato Vol. 8 No. 006 pp [] M.. Bot Mechacs of Icremetal Deformato Joh Wley ad Sos Ic. New York 965. [] M. M. Selm Reflecto of Plae Waves at Free Surface of a Itally Stressed Dsspatve Medum Recet dvaces Techologes Vol pp. 6-. [5] Y. C. Fug Foudato of Sold Mechacs Pretce Hall of Ida New Delh 965. Copyrght 0 ScRes. M