Computational Modeling for Acoustic Wave Propagation in a Layered Media

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1 Computatoal Modelg for Acoutc Wave Propagato a Laered Meda RAY R. ZHANG, AALHAMID ALAMIN Departmet of Mechacal Egeerg, College of Egeerg ad Computatoal Scece Colorado School of Me 1500 Illo St, Golde, CO UNITED STATES OF AMERICA rhag@me.edu, aalam@mmal.me.edu Abtract: - Th paper preet the of acoutc-emo (AE) wave propagato mult-laer materal ad mulato of AE wave repoe at free urface. I partcular, the AE ource modelled a a arbtraroretato dlocato over a cled-to-urface fault wth oe laer or at the laer-to-laer terface, whle the materal are aumed a mult-laer meda, each of whch homogeeou, otropc ad learl elatc. Wth the ue of the tegral traformato approach, the three-dmeoal wave propagato the materal olved traformed or frequec-waveumber doma. Subequetl a cloed-form oluto for wave repoe at free urface foud, whch ca the be coverted tme-pace doma. Numercal example are fall provded for llutrato. Ke-Word: - Acoutc emo, wave propagato ad catterg, dlocato ource 1 Itroducto Acoutc emo (AE) a aturall-occurrg pheomeo materal, whch the rapd releae of eerg from localed damage uch a crackg geerate elatc tre wave wth the materal. Aal of AE recordg at free urface ca help detect damage locato ad evert odetructve tetg ad evaluato for materal. Sce AE tetg ot reproducble due to the ource ature, exemplfed a udde ad ometme radom formato of a crack, modelg of AE ource ad wave propagato eetal. Sthe ad mulato of wave repoe wth the AE modelg have bee wdel ued tructural health motorg of uclear power, aeropace ad materal proceg [1-4]. The headg prevou reearch [5-7] ugget that AE tpcall codered a tme-depedet dlocato over a fte fault area, or mpl the fte dlocato ource. The fte dlocato ource ca be modeled a the ummato (or tegrato the lmt) of pot ource, each of whch accout for the evolutoar dlocato over a dcreted ub-area trggered at dfferet tme tat. Therefore, truthfull characterg the pot ource a ke udertadg of ature of the fte dlocato ource. The mecham of the above pot ource tpcall modeled a the product of a ource tme fucto characterg the dlocato growth, a factor combg e couple of mpulve force that equvalet to the ut dlocato, ad a calg factor or magtude (=fal dlocato x materal rgdt x fault area). Each couple (or dpole) ca be repreeted mathematcall b two mpulve force actg oppote drecto, wth a ftemal eparato dtace ether alog or perpedcular to the mpule drecto or, the lmt, b the dervatve of the mpule wth repect to the eparato-dtace parameter. The combato of the e couple are theoretcall equvalet to a kd of a dlocato (ormal ad hear) a arbtrar oretato from a elatodamc approach, whch ue the geeraled Bett recprocal relato b troducg a tme parameter, a frt decrbed [8]. Alteratvel ad more geerall Baku et al. [9-10] ue the cocept of tre glut to derve the ame reult. That dervato ue the aumpto that the dgeou ource are codered to be the reult of a localed, traet falure of the leared elatc cottutve relato, ad lead to the tre glut a a fucto of the dlocato quatt. Whle the above pot-ource characterato baed o equvalet force ueful olvg wave moto equato, whch have bee well developed ad wdel ued tud of emolog ad recetl fracture mechac, the ource decrpto ca be drectl gve wth a dlocato form ad ubequetl ued for fdg wave moto. I ISBN:

2 addto, the equvalet force are vald ol for decrbg a dlocato wth a medum, whch ot applcable to geeraled dlocato at the laer-to-laer terface, exemplfed a debodg damage realt. Bult upo the aforemetoed tude a well a pertet other [11-12], th paper preet a cotuum-mechac model for AE wave propagato laered materal, whch geerated b tme-depedet dlocato over a fte fault area bured wth oe of the laer or at the laerto-laer terface. whch j, k = x,, = 1,2,..., ad δ jk Kroecker delta, equal to oe f j=k ad ero otherwe, ad λ ad μ are the Lame elatc 2 2 cotat ad ca be foud a λ = ρ( α 2β ) 2 ad µ = ρβ term of P ad S wave peed (α ad β ) ad det ( ρ ) 2 Wave Propagato wth a Pot, Impulve Dlocato I th tud the AE ource decrbed a a mpulve dlocato rupturg at a pot wth a gve fault area, whle the materal are modeled a vertcall o-homogeeou or laered meda wth each laer beg otropc, homogeeou ad learl elatc, a chematcall how Fg. 1. The AE wave ca the be obtaed b olvg force-free wave equato of moto wth the dlocato. I dog o, t alo requred of pertet codto that are the cotut codto at each laer-to-laer terface ad boudar codto at both urface. For eae decrbg the oluto procedure, the followg dplacemet ad tracto vector are defed u, t) = u xex + u ex + u e (1) t, t) = τ e + τ e + τ e (2) where e x, e x x ad e are repectvel orthogoal ut vector the x, ad drecto, u the dplacemet, ad τ the tre. The wave moto ca the be foud b olvg the followg equato force-free laer τ = ρ u, (3) jx x + τ j + τ j, Where j = x,, = 1,2,..., ad ρ the det ad the prme ubcrpt deote the partal dervatve. The cottutve relatohp betwee dplacemet ad tree expreed a: τ = λ δ u + u + u ) + µ ( u + u ), (4) jk j ( x, x, j, k k, j j Fg. 1. Profle of laered materal wth bured dlocato ource. The cotut codto at each laer-to-laer terface requre that dplacemet ad tracto vector be cotuou acro = : u +, t) = u, t) ; (5) t +, t) = t, t) (6) + where ad repreet repectvel the egatve ad potve de of terface =. The boudar codto at oe free urface ad aother fxed or free oe are t = 0, t) = 0 ad t +, t) = 0 or u +, t) = 0 (7) (8) or boudar codto wth fte depth of laer,.e., radato codto that o propagatg wave come from the place where ft. ISBN:

3 The mpulve dlocato at a pot (x =0, =0, ) wth fault area A ca be decrbed a u( x +, +, +, t) u( x,,, t) = uδ ( t) (9) u δ ( t) = [ u e + u e + u e ] δ ( t) (10) Where u = [ u (coθ + u x ' x x' ] δ ( x) δ ( ) δ ( + θ ' ) ) (11) ad, = x,,.e., dlocato compoet u the global x-- coordate expreed term of ormal ad hear dlocato compoet ( u, u ) wth hear lp drecto ( θ ) local x - - coordate tem how Fg. 2. Fg. 2. Decrpto of dlocato a cled-tourface fault local x - - coordate 4 Numercal Example For llutrato, dplacemet wave repoe are computed at urface locato 1 (x=15 mm, =20 mm) ad 2 (x=30 mm, =40 mm) to a pot, mpulve hear-dlocato ource bured a uform half-pace of 7000-ere alumum allo [12]. The materal parameter ued ca be foud the ecod laer of Tab. 1, whle dlocato parameter are elected a u =0, u =0.01 mm, θ =0, ad fault area A=0.001 mm. For comparo, the dplacemet at the urface are alo computed for the three-laer medum gve Table. 1. Table 1. Phcal properte of laer α *10 km/m 3 β *10 km/m 3 Q α Q β ρ *10 kg/m 3 mm 1 4,32 2, ,32 3, ,32 4, Fgure 3 how the P-SV ad SH wave repoe ampltude at urface (.e., W S ad W T ) veru the wave umber radal drecto (k r ) at elected frequec 5M rad/. The domat peak of P-SV wave repoe (W S ) Fg. 3 for the uform half pace correpod to the propagatg P- ad S-wave mode at k r =ω/α=791 rad/m ad k r =ω/β=1572 rad/m, whle mlar peak for the three-laer medum are alo foud, howg the averaged propagatg P- ad S-wave mode over the three laer. No peak of SH wave repoe (W T ) oberved for the uform half-pace, dcatg that o propagatg SH-wave mode or urface wave ext for the uform half-pace. I cotrat, a couple of peak for SH wave repoe (W T ) how off for the three-laer medum Fg. 3, correpodg to the dfferet urface wave mode uch a Love wave. Fgure 4 depct the ampltude of x-drecto dplacemet the frequec doma for a uform ad a three-laer meda, dcatg more complext of wave repoe a laered medum tha the uform half pace addto to the mlar overall frequec feature. Fgure 5 how the x-drecto dplacemet at obervato locato 1 ad 2. It dcate that the dplacemet at locato 1 ero utl the frt P wave gal arrve at 4.0(10-6 ) whch cotet wth the theoretcal calculato baed o the P wave peed ad ource-to-repoe dtace. The S wave gal arrve later ad gve re to larger dplacemet ampltude tha that of the P wave at 8.2(10-6 ). The dplacemet at locato 2 how mlar wave propagato feature wth later arrval tme ad maller dampg-related ampltude. Fgure 6 ad 7 preet the x-drecto dplacemet ampltude the wave umber doma at elected frequece for a uform ad a laered meda, dcatg repectvel the combed P-SV ad SH wave repoe for the former ad ol P-SV wave repoe for the latter. ISBN:

4 M 0 m ω u x ω e ω t gd, t) = d dω π ( ) ~ (,, ; ) (12) 2 where M 0 the product of hear modulu, fal average lp ad lp area for hear dlocato, m(ω) the Fourer traformato of a ource tme fucto characterg the dlocato growth, ad u ~ d the dplacemet repoe to a ut-mpule, pot dlocato frequec doma. For the ource tme fucto elected a a ramp fucto wth the re tme T r, m(ω) gve b 1 ( (13) 2 ωtr ωτ m ω) = ( e 1) e ω Tr Fg. 3. P-SV ad SH wave repoe at 5M rad/ for a uform ad a laered meda. Fg. 4. The x-drecto dplacemet ampltude pectra a uform ad a three-laer meda Fg. 6. The x- drecto dplacemet ampltude the frequec-wave umber doma a uform medum. Fg. 5. The x-drecto dplacemet wave repoe at obervato locato 1 ad 2. The dplacemet wave repoe g d to a tmedepedet, pot hear dlocato ource ca be foud a Fg. 7. The x- drecto dplacemet ampltude the frequec-wave umber doma a laered medum ISBN:

5 The urface accelerato at locato 1 have alo bee computed for a laered medum, wth the ame materal properte a gve Table 1, a how Fg. 8 ad 9 tme ad frequec doma. Fall the x-drecto dplacemet repoe at obervato locato 1 are calculated wth the ame three-laer medum Table 2 but three dfferet dampg factor Table 2, whch how Fg. 10 ad 11. The clearl depct the fluece of the dampg rato wave repoe,.e., the maller the dampg factor, the larger the wave repoe. Fg. 8, accelerato v tme a three-laer medum. Fg. 10, Comparo of x-drecto dplacemet wave repoe at locato wth dfferet dampg factor frequec doma. Table 2: Comparo wth dfferet dampg factor (Q P ad Q S ) Dampg 1 Dampg 2 Dampg 3 P S P S P S Fg. 9, accelerato v frequec a three-laer medum. ISBN:

6 [3] M. Kalcka, Acoutc Emo a a Motorg Method Pretreed Cocrete Brdge Health Codto Evaluato. Joural of Acoutc Emo., 27, (2009), pp [4] L. M. Spaova, M. I. Ojova ad C. R. Scale, Acoutc emo techque appled for motorg ad pecto of cemettou tructure ecapulatg alumum. Joural of Acoutc Emo, 25, (2007), pp [5] L.B. Freud, Damc Fracture Mechac. Cambrdge Uvert Pre (1998). [6] A. Alam, R. Zhag, Modellg of acoutc emo ource ad wave repoe laered materal, Joural of theoretcal ad appled mechac, Sofa, vol. 44, No. 1, (2014), pp Fg. 11, Comparo of x-drecto dplacemet wave repoe at locato wth dfferet dampg factor tme doma. 4 Cocluo A cotuum-mechac model for AE wave propagato laered materal ha bee examed. The geeral dlocato ource modeled ad the correpodg tracto dcotut foud. Wave repoe olved aaltcall traformed doma. Some oberved wave feature are how wth umercal example. Further theoretcal vetgato, together wth umercal valdato of th tud wll be performed ad reported the ear future. Referece: [1] M. A. Hamtad, A Revew: Acoutc Emo, a Tool for Compote-Materal Stude. Expermetal Mechac, 26, (1986), pp [2] F. F. Baroum, J. Sulema, A. Korcak ad E. V. K. Hll, Acoutc Emo Motorg ad Fatgue Lfe Predcto Axall Loaded Notched Steel Specme. Joural of Acoutc Emo. Vol. 27, (2009), pp [7] C. U. Groe, ad M. Ohtu, Acoutc Emo Tetg - Bac for Reearch-Applcato Cvl Egeerg. Sprger Verlag, Berl & Hedelberg, Germa (2008). [8] R. Burrdge ad L. Kopoff, Bod force equvalet for emc dlocato. Bullet of the Semologcal Socet of Amerca, 54, (1964), pp [9] G. E. Backu ad M. Mulcah Momet teor ad other pheomeologcal decrpto of emc ource. I. Cotuou dplacemet, Geoph. J. Ro. Atro., Soc., 46, (1976a), pp [10] G. E. Backu ad M. Mulcah Momet teor ad other pheomeologcal decrpto of emc ource II. Dcotuou dplacemet, Geoph. J. Ro. Atro., Soc., 47, (1976b), pp [11] B. L. N. Keett, Semc wave propagato tratfed elatc meda. Cambrdge Uvert Pre (1983). [12] M.A. Hamtad, A.O. Gallagher ad J. Gar Effect of lateral plate dmeo o acoutc Emo gal from dpole ource. Joural of Acoutc Emo, 19, (2001), pp ISBN: