Diffraction of SH Waves by an Elliptic Inclusion with Partially Debonded Region in Bi-Material Half Space Ding xiaohao, Qi hui
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1 Diffractio of SH Waves by a Eiptic Icusio with Partiay Deboded Regio i Bi-Materia Haf Space Dig xiaohao, Qi hui Coege of erospace ad Civi Egieerig,Harbi Egieerig Uiversity,Harbi, Chia Keywords: bi-ateria haf space; SH waves; partiay deboded regio; cofora appig bstract: Base o eastodyaics, copex fuctio ethod with appig fuctio ad Gree s fuctio ethod are used to ivestigate the scatterig of SH waves by a eiptic icusio with a partiay deboded regio. The bi-ateria haf space is divided ito two parts aog the vertica iterface to ivestigate the two right age pae respectivey. Secody, iage ethod ad cofora appig ethod are epoyed to costruct the scatterig wave fied i part I. The the Gree's fuctio eeded is obtaied. Thirdy, with the aid of iterface cojuctio techique, a series of itegra equatios for deteriig the ukow force syste coud be set up through cotiuity coditios o the iterface ad Gree s fuctio. The ethod preseted i this paper ca be used to sove other siiar scatterig probe by arbitrary shape icusio with partiay deboded regio. Itroductio I the fied of coposite ateria, earthquake egieerig ad geotechica egieerig ad other fieds, deboded probes betwee eastic ateria are of theoretica ad egieerig appicatio vaue. Scatterig probes of SH waves [-7] as the sipest probes i the research of scatterig probe, have bee studied ay researchers, ad ay vauabe resuts have bee obtaied. However, the scatterig probes for soe specia structures are sti usoved.the bodig surface of differet aterias uder physica or cheica ijury wi occur the pheoeo debodig, which wi affect the echaica ad cheica properties. This paper aiy studies the effect o the echaica properties.may acadeics have studied the scatterig of SH waves by structures with deboded regio, but the atheatica ode estabished are aiy cocetrated i the whoe space ode or hoogeeous haf space. The scatterig of SH Mediu I Mediu II y C PDR r" b a o x x" Mediu III o waves by eiptic icusio with partiay deboded regio i bi-ateria haf space are rarey ivestigated. I this paper, copex fuctio ethod ad the idea of "cojuctio" are epeed to study the scatterig probes of SH waves by a eiptic icusio ear the vertica iterface i bi-ateria haf space.iage ethod ad cofora appig ethod are used to study the wave fied outside the eiptic icusio. I the eiptic doai, there are stadig waves that satisfy the cotiuity coditios of dispaceet ad stress o the boudary of doai I ad doai III. d the stress cotiuity coditio shoud be satisfied o the deboded regio. Fig. shows a bi-ateria haf-space with a eiptica icusio with partiay dedodeb regio (PDR of haf-acroaxis a ad haf-brachyaxis b. The eiptic icusio has ateria properties, shear oduus 3 ad ass body desity 3. The distaces fro the ceter of eiptica icusio to the vertica iterface ad horizota surface are d ad h, respectivey. The icidet age of SH waves is. There are two coordiate systes settig o O ad O poit. h C d Fig. Eiptic icusio with partiay deboded regio i bi-ateria haf space y" 6. The authors Pubished by tatis Press 89
2 Basic theory Cojuctio ethod wi be used i this paper, Gree s fuctio shoud be studied before. Gree s fuctio used i this paper is a essetia soutio of dispaceet fied for a quarter-pae with a eiptica eastic icusio by ati-pae haroic ie source ( z z ( z x iy oadig at vertica surface. The dispaceet fuctio G ust satisfy the goverig equatio with oitted the tie haroic factor exp( - it of the foowig for: G kg z z 4 ( I which, z ad z are the copex variabes, k / cs is wave uber, ad cs = / are the disturbig circuar frequecy ad the shear veocity of the edia, ad are the ass body desity ad the shear oduus of the edia, respectivey. The stresses correspodig to Eq.( ca be writte as: G i G i G i G i rz ( e e, z i ( e e ( z z z z For wave scatterig probes ivovig eiptica icusio i the copex (z, z pae, it is possibe to ap the itera/extera regio of the eiptica icusio ito the iside/outside regio of the circe(i the (, pae. Itroducig the appig fuctio: i Z ( R(, Re (3 The above appig fuctio ap the outside of the icusio i the (z, z pae ito the regio η. Cosequety, the correspodig goverig Eq.( i (, pae takes o the foowig for: (4 G kg ( ( 4 I (, pae, Eq.( ca be writte as: G G i G G r ( ( R (, R ( (5 ti-pae SH waves i doai I ad doai II Cosiderig the existece of the free boudary ad the iterface i bi-ateria edia, the iage ethod is epoyed to trasfor right age space to fu space. we ca cosider the fu space ode as a equivaet ode with utipe wave sources. d the equivaet icidet wave ca be expressed as foows: (, ie ik i i i W W exp{ [( ( i he ( ( i he +( ( ih de (6 i ( ( ih d e ]} Where, π α, ad W are icidet age ad the apitude of the icidet wave respectivey. Siiary, the equivaet refected wave ad refracted wave are: ( r,e ik i i i W W exp{ [( ( ih e ( ( i h e ( ( ih d e (7 i ( ( ih d e ]} 6. The authors Pubished by tatis Press 9
3 ( f,e ik W W exp{ [( ( i he ( ( i he ( ( ih de i ( ( ih d e ]} i i i I which, π α, γ π α, α ad α are the refectio age ad refractio age respectivey. (8 Stadig waves i doai III Costructig stadig wave i the eiptic doai eed satisfy the stress free coditios o boudary C ad cotiuity coditio of dispaceet ad stress o boudary C :, ( C (st rz 3kG 3 ( [ ( 3 ( ( 3 ( ][, ( ] (9 C J k J k ( C I which, C is ukow coefficiet, W T is dispaceet apitude of stadig wave, W T = Stadig waves i eiptic doai: (st ( W WT D J ( k3 ( [ ] ( ( I which, D is ukow coefficiet. The Fourier expasio of Eq.( withi the rage [, ] ca be writte as (st 3kW 3 T ( rz Ca [ J ( k3 ( J ( k3 ( ][ ] ( ( I which: a e e i( i( i(,, I coordiate (,, ad are the start age ad ed age of deboded regio. Coparig Eq. ( with the stress of Eq. ( uder the coditio J k ( J k ( D G C a ( 3 3 (3 J k3 ( J k3 ( The stadig wave fied i doai II ca be expressed as: J k ( J k ( ( 3 3 (st G W C aj( k ( [ ] (4 J ( k3 ( J k3 ( The cotiuity coditio o the boudary of doai III ad doai I (i,e ( re, (s (st W W W W, C (5 ( i,e (r,e s (st rz rz rz rz, (, The Gree fuctio we eed ca be get by soved Eq.(5. 6. The authors Pubished by tatis Press 9
4 s show i Fig., ukow force systes f ad f are oaded o the cojuctio sectio to satisfy the cotiuity coditios o the iterface. So a series of Fredho itegra equatios for deteriig the ukow forces ca be set up. (I (II The tota dispaceets W, W ad tota (I (II stresses z, z i the two parts are: (I ( i,e ( r,e ( s (II ( f,e W W W W, W W (6 (I ( i, e ( r, e ( s (II ( f, e z z z z, z z Fig. Cojuctio ode (s Where, W is the dispaceet of the scatterig (s wave. z is the stress of the scatterig wave. The stress cotiuity coditio ca be expressed as: (I (II z si f( r, z si f( r, (7 Where r ad are the poar coordiates i the goba coordiate syste x o y ad z r exp( iθ, z z d. whe = /, r, whe = /, r h. ( i, e ( r, e ( f, e ccordig to z z z, we ca get: f( r, f( r,,, (8 The dispaceet cotiuity coditios ca be writte as: ( I ( f ( c (II ( f ( c W W W W W W (9 ( i,e ( r,e ( f,e ccordig to W W W, we ca obtai: ( s ( f ( f W W W ( ( f Where, W is the dispaceet fied caused by force syste f, ad ( f W is the dispaceet fied caused by force syste f. ccordig to the cotiuity coditio ad the Gree s fuctio we have obtaied, the itegra equatios with ukow ati-pae forces ca be expressed as: f ( r, [ G( r, ; r, G ( r, ; r, ] dr (S f( r, [ G( r, ; r, G( r, ; r, ] dr [ W ] f( r, [ G( r, ; r, G( r, ; r, ] dr (S f( r, [ G( r, ; r, G( r, ; r, ] dr [ W ] I which, G ad G are the Gree s fuctios i doai I ad II respectivey. Hoop stress aroud the icusio ca be expresses as ( (I G( r, ; r, G( r, ; r, z z f( r, dr f( r, dr r r ( I ( i,e ( r,e ( s I which, z z z z ( ( (3 Cocusio I this paper, oy theory deductio is preseted, aaitica resuts wi be give i the foow-up work. Refereces [] Lee V W, Maoogia M E. Surface otio above a arbitrary shape udergroud cavity for icidet SH wave[j]. Europea Earthquake Egieerig, 995, 8(: The authors Pubished by tatis Press 9
5 [] Lee V W, orwogpaibu. Scatterig of ati-pae (SH waves by a sei-eiptica hi: I Shaow hi[j]. Soi Dyaics ad Earthquake Egieerig, 3, 53: 6-5 [3] LUO Hao, V W Lee, LING Jia-we. ti-pae(sh waves diffractio by a udergroud sei-circuar cavity: aaytica soutio[j]. Earthquake egieerig ad egieerig vibratio,,9(3: [4] QI Hui, YNG Jie. Dyaic aaysis for circuar icusio of arbitrary positios ear iterfacia crack ipacted by SH-wave i haf-space[j]. Europea Joura of Mechaics / Soids,, 36: 8-4. [5] QI Hui,YNG Jie,SHI Yog,TIN Jia-yog. Dyaic aysis for Circuar Icusio ear Iterfacia Crack Ipacted by SH-Wave i Haf Space[J]. Joura of Mechaics,, 8(: [6] YNG Zai-i, YN Peiei, LIU Dia-kui. Scatterig of SH-waves ad groud otio by a eastic cyidrica icusio ad a crack i haf space[j]. Chiese Joura of Theoretica ad ppied Mechaics, 9, 4(: 9-35.(i Chiese [7] Jiaxi Zhao, Hui Qi, Diakui Lu, Hogiag Li. Scatterig of SHwaves by a shoow buried circuar ier with a partiay deboded regio[j]. CHINESE JOU RNL OF SOLID MECHNICS, 8,9(3:3-36(i Chiese 6. The authors Pubished by tatis Press 93
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