Three-Dimensional Defect in a Plate Boundary Element Modeling for Guided Wave Scattering

Transcription

1 Ky Egg Matals Vols (004 pp ol at Tas Th Pblatos, Swtzlad Ctato & Copyght (to b std by th pblsh Th-Dsoal Dft a Plat Boday Elt Modlg fo Gdd Wav Sattg Xaolag Zhao, Josph L. Ros Itllgt Atoato, I., 75 Stadsh Pla, St 00, Rovll, MD 055, USA Dpatt of Egg S ad Mhas, Th Psylvaa Stat Uvsty, Eath-Egg S Bldg, Uvsty Pa, PA 60, USA Kywods: Ultaso Gdd Wavs, Boday Elt Mthods, Wav Sattg Abstat. A hybd th-dsoal boday lt oal od paso thq s dvlopd to stdy th gdd wav tg fo a abtay shap dft a plat. Lab wav dt to a la thogh plat hol was stdd as a apl. Th td wav dsplat fld ad ts oal od opot a gv. Mod ovso fo Lab wav to S wav at th dft boday w obsvd. Itodto Ultaso gdd wavs hav dostatd gat pottal fo dttg ay ds of dfts that o tbs, pps o plat stts. owv, dft lassfato ad szg by sg gdd wavs s stll a dfflt pobl to b solvd d to th oplty of wav popagato haatsts. Aalytal stds of gdd wav tg fo a dft had ltd sss fo dfts of spl gots [, ]. Nal tools a bog absoltly sstal to stdy th gdd wav tg pobls opl-shapd wavgds o dfts. Ral wold dfts a th dsos, ad th dft wdths a oft aow tha th spto wav ba-wdth. Thfo, a th-dsoal (3-D odl of gdd wavs tg fo thos dfts s qd. So po wo ths fld a b fod [3-7]. Boday Elt Mthod (BEM, as o of th gg aalyss thqs, has th advatags sh as dto of th dsoalty, lss opt t ad stoag, asy aagt of bodd doas, ad o tagtd allatos ov th popla Ft Elt Mthod (FEM. Th Noal Mod Epaso (NME thq has also b sd wav fld aalyss ad sythss. I ths pap, w pst a hybd odl of th dsoal (3-D BEM obd wth th NME thq to allat th gdd wav tg fld fo a 3-D dft a plat. Th agla pofl of th dsplat fld as wll as ts gdd wav od opot, td fo a la thogh plat hol wth S0 od Lab wav d, s pstd. Th od ovso phoo fo Lab wav to S wav was also obsvd. Lsd to P Stat Uvsty - Uvsty Pa - USAJosph L. Ros (JLRESM@g.ps.d - P Stat Uvsty - USA All ghts svd. No pat of th otts of ths pap ay b podd o tasttd ay fo o by ay as wthot th

2 454 Advas Nodsttv Evalato Ttl of Pblato (to b std by th pblsh Thoy θ Idt gdd wav Fg.. A shat pobl of a pla gdd wav dt to a th-dsoal dft a ft plat. Pobl Stp ad Copt of th Appoah Cosd a 3-D gdd wav tg pobl as show Fg., wh a ftly lag flat plat va has a 3-D abtay shapd dft (.g. a la thogh plat hol. A t-hao gdd pla wav (th Lab wav o S wav s dt th postv dto. Th gd wav od dt to th dfts wll slt td wavs of all ods of th popagatg as wll as o-popagatg ods (both Lab wav ad S wavs at th vty of th dft. Th vast ods d ot qly wth sval wavlgths; oly popagatg ods svv ad ay ot gy fl at th fa-fld fo th dft []. At dfft dtos, th wav dsplat ad stss flds a dfft gal ad fo a agla patt. Fo dfft dfts, ths patt wll also b dfft, whh old t b sd fo dft shap ad szg aalyss. I od to allat th td wav fld, th hybd BEM oal od paso thq that was sssflly appld to th -D dft haatzato pobl [] was tdd to th 3-D ass. I spf, a vtal yld losg th go wth dfts th plat was dfd fo BEM shg. Th ads of th ds go s hos to b lag tha th lagst wavlgth of all possbl popagatg gdd wav ods. A yldal oodat syst was stablshd at th t of th ds go, wth th z0 pla odg wth th d-pla of th plat. Otsd th dfd ds go, th total dsplat ad tato fld a th spposto of all th dt ad td popagatg wav ods. Wth th dft go, th 3-D lastodya BEM was sd to allat th wav fld. O th boday odtos a spfd at th vtal yldal boday, th BEM allats ad otpts th dsplat ad tato val fo ah boday lt, fo whh th td wav fld s dtd ad so as ah wav od opot. Th Dsoal Gdd Wav Noal Mod Epaso Th Lab wav ad S wavs a th dsoal ft plat a b pssd th fo [3, 5] θ z V V W φ (, θ φ (, θ (Lab wav θ φ(, θ ψ U (, θ l θ ψ θ U (, θ (S wav ( l 0 z

3 Ky Egg Matals Vols Ttl of Pblato (to b std by th pblsh wh s th th od Lab wav b ad a b dtd by th -D Raylgh-Lab wav dspso qatos; l s th wav b of th od S wav. V ad W a th Lab odal fto th plat thss dto as th -D ass; U (z s th S od shaps. Th pottal ftos φ(, θ ad ψ (, θ satsfs lholtz qato,..: φ φ φ φ 0, ( θ ad ψ ψ ψ l ψ 0. (3 θ Cosqtly, soltos of th pottals ftos hav to b fod od to dsb th wav fld. Thy a b gv th followg fo φ (, θ Φ( ψ (, θ Ψ( θ, θ, fo Lab ods fo S ods (4 wh Φ ( ad Ψ ( a th soltos of Bssl s dfftal qatos,.. d Φ dφ + + ( d d d Ψ dψ + + ( l d d Φ 0, Φ 0, fo Lab ods fo S ods (5 Th stabl soltos to qatos (5 fo th td wav fld a b pssd as al ftos of th fst d,.. Φ ( (, Ψ ( l ( l (6 Ths th td wav dsplat fld a b pssd as: N L θ AV '( B U ( l l θ (7 ( θ θ B U '( l ( N L θ AV 0 0 z N 0 A W ( θ ( wh th offts A ad B lat to th otbto fo th Lab ods ad S ods, sptvly. N ad L a th total b of Lab ad S ods that a popagat th plat. Th lvat stsss fo th td wav fld a also b padd by sg oo s law,

4 456 Advas Nodsttv Evalato σ σ θ N 0 + N L A 0 0 L 0 B [ Σ B A Ttl of Pblato (to b std by th pblsh Σ ~ ( Σ µ U [ θ [ µ U [ [ ( l l ' ( l + ( l ' ( '( l ( l ] θ ( ] θ + ( l ] θ ( ] θ (0 ( σ N L θ θ z A Σz '( + µ BU '( z ( l ( 0 0 l wh ftos Σ (z a th offt latd to th wav stt of th th od. Th total dsplat ad stss fld th plat a atally th s of th og ad td flds,.. + ; σ + σ σ. (3 Th-Dsoal BEM Folato of th Elastodya Pobl Th boday tgal folato fo th t-hao lastodya stat wth zo body fo, dfd ov th go Ω wth boday s p d p d (4 whh th dsplat opots ad tato dsplat at ay boday pot a latd to th dsplat p opots ov th whol boday, ad th fdatal solto ad p o th boday d to a t load at pot. Th offt dpds o th soothss of th boday f pot s a boday pot. δ / fo sooth boday pots ad δ fo tal pots. Aft dstzg th boday of th ds go to NE lts. Eqato (4 a b tasfod to NE + p d d p (5 NE vto oto s sd,.. p ; p p ; 3 p 3 By dfg 3 p p p3 3 ; p p p p3 (6 3 3 p3 p3 p G d, p d Th syst qato fo od bos, ad + wh wh

5 NE NE Ky Egg Matals Vols Ttl of Pblato (to b std by th pblsh G p (7 o a at fo U G P ( Th fdatal soltos fo a th-dsoal lastodya pobl a b wtt as [0] p [ Uδ U 4πµ, l, ] du [( U ( δ +, l U (, l l, 4π d du L du du, l, + ( ( U, ] d d d T ( (0 wh U U p( p( p( + ( + ( p( ( 3 p( + + ( ( ( ad L ; T. Cosd th sh goty as show Fg. ad spaat th boday lts fo gops Top, Cyl, Bot ad Df, whh stads fo th lts o th top plat sfa (lds th dft, vtal yldal boday, botto sfa ad thos o th dft. Th qato (3 a b wtt as Top 0 tcyl (3 Bot 0 Df 0 [ ] Cyl [ G] th tato f boday odtos a alady pltd to th qatos. Th oly o-zo tato t s fo th vtal yldal boday, wh th dt wav ad td wavs sppos. Ass that th a lts th z dto ad l lts th θ dto. Apply qato (3 to ah od, ad tat th Fo tasfo to t M (stad of. Th dsplat ad tato at o th vtal boday a b wtt as I BS { } 3 l [ ] 3 l + [ ] 3 l JV { A} JV (4 I BS { } 3 l [ t] 3 l + [ t ] 3 l JV { A} JV t (5

6 45 Advas Nodsttv Evalato Ttl of Pblato (to b std by th pblsh wh J s th a b of ods that a popagat th plat, ad V s th tato paat fo th agla fqy,.. J N + L +, V M +. [ ] BS 3 l JV ad [ t ] BS 3 l JV a th g od wav stt at whh ah ol psts o g od ad ah ow gvs o opot of th dsplat o tato at o vtal lt. Elat { A } JV qato (4 ad (5, w a gt I BS BS I BS BS { t} [ ] [ ] [ ] [ ] [ ] [ ] { } 3l t 3l t 3l JV JV 3l 3l + t 3l JV JV 3l 3l, (6 ths qato (3, t Cyl a b pssd as Cyl, whh t a b solvd fo th la algba qato. O Cyl s obtad, th td wav dsplat fld ad tato fld a b allatd basd o qato (3. Ftho, th tg offts fo ah wav od a b optd basd o qatos (4, (7 ~ (. To h th valdty of th BEM solto, th gy osvato thod was sd. It stats that a losd wav fld go wth o so o s, th wav gy flows to th go shold b qal to that flows ot. Th avag pow P adatd aoss a losd sfa S a b valatd as [] ( ( ( σ( ω P I ds, (7 s wh s th oal vto of th sfa S ad th sybol I dots th agay pat of a opl b. Nal Rslts Cosd a al pt as shatally show Fg., wh th plat thss s, th ads of th thogh-plat hol s, ad th ads of th vtal yldal boday s 0. Th a 00 lts foly shd o th vtal boday. Th total b of sh lts s aod 00. Cosd a pla S 0 od Lab wav of fqy 0. Mz popagatg th postv dto. Th allatd, θ ad z opot of th td wav patl dsplat apltd o th vtal yldal sfa s show Fg. (a, (b ad (, sptvly. Fg. 3 (a, (b ad ( shows th odl doposto vals fo th S 0, A 0 ad S 0 od of dfft agla Fo opots. I ths spf as, w tatd th agla paso ts to M stad of ad dsadd hgh od ts s thy a glgbl apltd, so -M,,M opots a dsplayd fo to 5 thos fgs (a (b ( Fg.. (a opot, (bθ opot, ad ( z opot of th td wav dsplat apltd o th vtal yldal boday.

7 Ky Egg Matals Vols Ttl of Pblato (to b std by th pblsh It s s that all th th dsplat opots a syt wth sptv to th as. Nal pts wth vaos bs of lts fo sh gato ad dfft sz vtal la go showd sla t pofl patts, whh dat good stablty ad ovg of th od dvlopd. Th al o fo th allato fo qato (7 s fod aod.5%. Fo Fg. 3 (a, (b ad (, w old s that th td wav has two ao opots, o s th S 0 od Lab wav ad th oth s th 0 od S wav. It s ths la that od ovso os at th la thogh plat hol fo th S 0 od Lab wav to 0 od S wav, bt ot to A 0 od Lab wav althogh all ad oly ths th ods a adssbl at that fqy. Fg 4 (a ad (b daw th agla apltd pofl fo th dsplat of S 0 od Lab wav ad 0 od S wav. Not that thos fgs, th wav vto dto s ot ssaly th adal dto d to th goty of th dft ad possbl wav gatg aod th thogh-plat hol. Nothlss, th td Lab wav apltd s s azd alog th as ad th S wav s aod 45 dgs away fo gatv dto (a (b ( Fg. 3. (a S 0 od Lab wav, (b A 0 od Lab wav, ad ( 0 od S wav agla Fo opot apltd of th td wav patl dsplat o th vtal yldal boday Fg. 4. Th td (a S 0 od Lab wav ad (b 0 od S wav patl dsplat apltd o th vtal yldal boday

8 460 Advas Nodsttv Evalato Colsos Ttl of Pblato (to b std by th pblsh ybd boday lt oal od paso thq was sd to odl th gdd wav tg fo a 3-D dft. Th dvlopd od was appld to stdy th td wav fld fo a la thogh-plat hol wth a pla S 0 od Lab wav d. Th td dsplat fld was allatd ad plottd apltd agla pofl. Modl doposto of th td wav fld gav th apltd of ah wav od that was adssbl th plat. Mod ovso was obsvd fo Lab wav to th S wavs at th thogh-plat hol. Aowldgts Ths wo s patally fdd by th Oa Rdg Natoal Lab, Oa Rdg, TN. Rfs [] Rohl, S., J. Aost. So. A. 67, (0. [] Wag, X. ad Yg, C. F., J. Aost. So. A. 0, (00. [3] Ahbah, J. D. ad X, Y., J. Aost. So. A. 06, 3-0 (. [4] Chag, Z. ad Mal, A., Mhas of Matals 3, 7-04 (. [5] Dlgt, O., Gah, T., Bosto, A., Cawly, P. ad Low, M., J. Aost. So. A., 5-60 (00. [6] Fo, P. ad Say, M. B., J. Aost. So. A., (00. [7] Gah, T., Wav Moto (003. [] Vasdva, N. ad Mal, A. K., J. Appl. Mh. 07, 6- (5. [] Zhao, X. ad Ros, J. L., It. J. Sol. Stt., (003. [0] Dogz, J., Boday Elts Dyas, Coptatoal Mhas Pblatos, Sothapto Bosto 3. [] Vaathaal, V., J. Math. Phys., 5-5 (77.