Manipulate Elastic Wave Modes. by an Ultrathin Three-component Elastic Metasurface

Transcription

1 Mnipulte lsti Wve Modes y n Ultrthin Three-omponent lsti Metsurfe Pi Peng*, Cheng Feng nd Kngheng Zhou Shool of Mthemtis nd Physis, Chin University of Geosienes, Wuhn , Chin Astrt We design two-dimensionl ultr-thin elsti metsurfe onsisting of steel ores oted with elliptil ruers emedded in epoxy mtrix, ple of mnipulting ulk elsti wve modes for refleted wves. The energy exhnges etween the longitudinl nd trnsverse modes re ompletely ontrolled y the inlined ngle of ruer. One elsti mode n totlly onvert into nother y the ultr-thin elsti metsurfe. The onversion mehnism sed on the nondegenerte dipolr resonne is generl method nd esily extended to three-dimensionl or mehnil systems. A mss-spring model is proposed nd well desrie the onversion properties. We further demonstrte tht high onversion rtes (more thn 95%) n e hieved stedily for one elsti metsurfe working on lmost ll different solid kgrounds. t will ring wide potentil pplitions in elsti devies. -mil: pipeng@ug.edu.n

2 n the pst dede, the emerging of metsurfes [1], whih re ultrthin mteril sls with su-wvelength rtifiil strutures providing grdient phse dely on the surfe, mke it possile to modulte eletromgneti wvefront. The generlized Snell s lw opens new degree of freedom in mnipulting eletromgneti wves. This onept of metsurfes hs een extended to ousti metsurfes. Full ontrol (phse nd mplitude) of slr ousti wves y ousti metsurfes hs ttrted lot of reserh interests [-10]. Vrious of ousti metsurfe sed on oiling up strutures [-4] or Helmholtz resontors [5] re proposed to modulte ousti wvefront. Very reently, Ghffrivrdvgh et l. [7] nd Zhu et l. [8] present ousti metsurfes enling ousti ontrol with simultneous phse nd mplitude modultion [9]. These works on mplitude mnipultion fill the gp in full ontrol of ousti wves. lsti wves exhiit rih polriztion thresti sent in ousti nd eletromgneti wves. Beside the phse nd mplitude, polriztion modultion is nother hllenging projet speilly for elsti wves. The onept of metsurfe provide new wy for modes onversion of elsti wves. Until now, the modulting of elsti wve wvefront y elsti metsurfes is widely studied in solid sls [11-16]. Very few works re rried out on ulk elsti wves [17-0]. To the est of our knowledge, it is still lnk in of mplitude or polriztion properties mnipultion for elsti wves. Some efforts hve een mde on interesting mode-oupled wve ehvior in solids [0-3]. However, ritil inident ngles or nisotropi mterils re used in hieving totl wve onversions. The mnipultion of elsti wve modes y elsti metsurfes hs not yet een explored. n this letter, we design n ultr-thin elsti metsurfe sed on three-omponent resontors [4,5], whih feture the elsti metsurfe t low frequeny. The elsti metsurfe

3 rings strong ouplings etween nd T modes, resulting in high mode onversions. We fous on the onversions from -to-t modes, euse it is hrder thn the onversion from T-to- modes, nd T wves hve gret potentil in medil nd industril pplitions due to its short wvelength [1]. We onsider n elsti metsurfe pled on the surfe of semi-infinite epoxy kground, s shown in Figs. 1(). The elsti metsurfe onsists of periodi steel ylinders oted with elliptil soft ruers emedded in epoxy mtrix, s shown in Figs. 1(). The rdius of the steel is r= 0. p, where p is the periodi onstnt. The semimjor nd semiminor xes of the elliptil ruer lyer re r = 0.38 p nd r = 0.4 p, respetively, nd is the inluded ngle etween the mjor xes of ellipse nd the horizontl diretion (x-xes). The prmeters of mterils used re: 3 e = 1180kg/m, l,e 540m/s = nd t,e = 1160m/s for epoxy; 3 r = 1300kg/m, l,r 55m/s = nd t,r = m/s for ruer [4]; 3 s = 7900kg/m, = 5800m/s nd t,s = 300m/s for steel, respetively, where is the mss density, nd l,s l nd t re nd T wve speeds. We let plne wves normlly inident from ottom, nd oth nd T modes re found in refleted wves. Tht mens fter refleting y the elsti metsurfe, some inident wves hve een onverted into T wves. The -to-t modes onversion rtes (C) n e desried y the rtio T / of their energy flux long the y-diretion: p * ( v ) dx v dx T 0 x xy = p * ( ) 0 y yy (1), where v nd re the veloity (symol * mens onjugte) nd stress on the interfe etween the elsti metsurfe nd the epoxy kground, nd sripts () nd T () denote the refleted

4 (input) wve nd trnsverse (longitudinl) mode, respetively. The C s funtion of frequeny nd the inluded ngle is omputed y COMSO Multiphysis softwre sed on the finiteelement method. As shown in Figs. 1(), the vlue of C is generlly very smll due to the wek oupling etween nd T modes. rge Cs re found ner two frequenies ( t,e / ) or ( t,e / p) p T mximum vlue ( / 1 or 0.19( ). n prtiulr, C n reh its theoretil due to the onservtion of energy) t est-ngle = 0.31( ) = for the frequeny or, respetively. Tht mens there re no wves in refletion. The inident mode elsti wves hve een totlly onverted into T mode in refleted wves. The two white thik rrows shown in Figs. 1() point out where the totl onversion effet ours. We note tht the wvelength in epoxy kground is out orders higher thn the thikness of the elsti metsurfe. We further lulte the nd struture of n elsti metmteril mde of the sme squre unit ells s the elsti metsurfe. The inluded ngle is =. The nd struture of the elsti metmteril is plotted in Figs. (). The lowest flt nd t frequeny ( t,e / p) is indued y lolly rottionl resonne [6]. The seond nd third rnh re tully indued y non-degenerte dipolr resonne [5]. The displement field distriutions of the eigensttes of the eigenmodes on the seond nd third rnh t the X point of the Brillouin zone re plotted in Figs. () nd (), respetively. The movements of the ores in these two eigensttes re perpendiulr to eh other. The eigenfrequenies of the two non-degenerte eigenmodes re ( t,e / ) nd ( t,e / p) p, whih re extly the frequenies t where the totl onversion is otined. These two eigenfrequenies re lmost independent with the inluded ngle.

5 The onversion mehnism n e explined with simple piture. n intuitively, the high onversion must rely on two ftors: the elliptil ruer lyer nd the resonne inside the elsti metsurfe. The inlined elliptil ruer lyer reks the system symmetry nd mkes the nd T modes oupled to eh other. The oupling is wek unless the resonne is involved. When frequeny is ner to the eigenfrequenies or, the exited dipolr resonne rings very strong olique virtions inside the elsti metsurfe. To stisfy the ontinuous oundry ondition on the interfe etween the elsti metsurfe nd the epoxy kground, T modes is required to exist in refleted wves if horizontl movements re sent in inident wves. As result, -to-t modes onversion is oserved. The T-to- mode onversion n e otined in similr wy. The design of dipolr resonne indued modes onversion is esily extended to three-dimensionl or mehnil systems. We propose mss-spring model, s shown in Figs. 3(), to ptures the min physil essene of the onversion effet nd revels the reltion etween the C nd the inluded ngle. The model onsists of resontor [5] onnets to hlf one-dimensionl montomi hin. The elsti metsurfe is modeled s the resontor. The ore nd mtrix in one unit ell re modeled s mss m (denoted y red disk) nd M (denoted y green ring), respetively. The mss m is onneted y four springs to the mss M. The stiffnesses of the two springs, whih is indued y the deformtions inside the ruer lyer, long the mjor xes diretion of the ellipse re given y nd G for extensionl nd sher displements [5], respetively. Similrly, the stiffness of the two springs long the minor xes diretion of the ellipse re given y nd G, respetively. The horizontl (vertil) displements of mss m nd M re u ( v ) nd U ( V ), respetively. The expressions of the fores F m, nd F m, exerted on m long the mjor nd

6 minor xes diretions of the ellipse, respetively, re given s following: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Fm, = u U os + v V sin G u U os + v V sin Fm, = v V os u U sin G v V os u U sin (). As plne wves re normlly inident to the elsti metsurfe, the semi-infinite solid kground is modeled s hlf one-dimensionl montomi hin. The mss of the hin is m 0 relted to the mss density of the epoxy kground, nd the spring stiffness of the hin is nd G relted to the ompressionl nd sher modulus of the epoxy kground, respetively. The distne etween neighor msses m 0 is denoted s d. The horizontl nd vertil displement of the n-th mss m 0 is denoted s u n nd v n, respetively. We pply the ssumptions of hrmoni time dependene i t e iqd nd the Bloh s ondition v = n+ 1 vne ( iqt d u = n+ 1 une ) to the hin, where q q = / l,e ( T t,e = / ) is the Bloh wve vetor for longitudinl (trnsverse) wves. The displements on the 0-th nd 1-th mss m 0 n e written s: v0 = A + A u0 = B + B = + iqd iqd v 1 = Ae + Ae iqtd iqtd u 1 Be Be (3). Here A ( B ) nd A ( B ) re ritrry mplitude for upwrd nd downwrd (T) wves. The 0-th mss m 0 is onneted to the mss M y spring the sme to the springs etween the msses m 0. Aording to Newton's seond lw, we n get the equtions of motion in glol oordinte:

7 mu = Fm, os Fm, sin mv = Fm, sin + Fm, os MU = G ( U u ) Fm os + F MV = ( V v ) Fm sin F m0u0 = G ( u0 U ) G ( u0 u 1 ) m0v0 = ( v0 V ) ( v0 v 1) 0, m, 0, m, sin os n se of wve inidene, we hve B = 0. By using qs. () nd qs. (3) into qs. (4), there (4). re six equtions for six unknows s u, v, U, V, A, nd B. The expression of C n e T otined from the model s ( t,e l,e ) / = / B / A. The nlytil results of the C re plotted in Figs. 3(), whih is greed well with the simultion results shown in Figs. (). The totl onversion effets re otined t the sme frequenies nd the sme inluded ngles. The prmeters used in the model re m = s r, M e ( p rr ) =, m 0 = ed, = ( + G ) / m, ( ) = + G / m, =, e l,e G= nd d = p, where the e t,e vlue of nd re otined from the nd struture shown in Figs. (). The expression of C n e gretly redued thus giving ler piture for the onversion effet under two pproximtions: "low frequeny" nd "soft ruer". The studied system (shown in Figs. 1()) is physilly qulified to meet these two pproximtions. As the dipolr resonne re set t very low frequeny, resulting from hevy ores nd soft oting lyers, n e regrded s n infinitely smll quntity. As the ruer is muh softer thn the other mterils, the terms with modulus rtio etween the ruer nd other mterils / (or G / G, et.) re lso regrded s infinitely smll quntities. After ignoring the high-order terms of the smll quntities, the expression of C n e rewritten s:

8 T sin ( ) t,e = (5). l,e t,e os + sin sin os + + l,e qution (5) shows tht the vlue of C stritly equl to zero in two onditions: sin = 0 or = 0. The first one mens the elliptil ruer lyer is orthogonlly pled, nd the seond one mens ruer lyer is irle not ellipse. Both these two onditions led to deoupling etween the nd T modes. To otin high C, we let the frequeny equl to redued to, then the q. (5) is T t,e l,e = (6). tn + l,e t,e l,e ot At the resonne frequeny, the C is s funtion of the inluded ngle. nequlity operting on the denomintor on the right side of the eqution, q. (6) is rewritten to T t,e l,e = 1 (7). l,e t,e l,e The inequlity sign shows tht the vlue of C n reh to one iff t,e tn = l,e ot, whih gives the expression of est-ngle rtn l,e / t,e =. To our surprise, the est-ngle is independent with the elsti metsurfe ut simply determined y the rtio etween nd T wve speeds (or sy the Poisson s rtio) in the solid kground. We plot the nlytil results (red line) of C otined from q. (6), whih perfetly predit the simulted results (lk dots), showing in Figs. 4(). The refleted wve will e pure wve when equls to 0 or /, where no onversion hppens. On the ontrry, the refleted wve will e pure T wve when the equls to, where the totl onversion hppens. Otherwise, oth nd T modes will e found in refleted wves. The ompositions of elsti modes in refleted wves re ontrollle independently y hnge the inluded ngle. Similrly, y using = into q. (5), we get

9 = rtn / =. As shown in Figs. 4(), the simulted results (lue squres) re t,e l,e lso perfetly predited y the model (green line). Another interesting phenomenon is tht the high onversion effet is very stedy for different T kground mterils. As shown in Figs. 4() red line, high C (defined s / 0.95 ) is hieved in rnge of from 0.8 to The orresponding rnge of the rtio etween nd T wve speeds is from to 3.69 (Poisson s rtio from 0 to 0.46). Atully, it is n extremely wide rnge overed lmost ll the solid mterils. For exmple, when the epoxy kground is repled y eryllium, whih is ommon mteril with lrge differene etween nd T wve speeds, the hnge of the est-ngle is very smll. For fixed inluded ngle =, the C will only slightly redue from 1 to 0.98 fter hnging the kground, s shown in Figs. 4(). For other ommon solids suh s luminum, glss, or led insted of eryllium, the Cs keep higher thn We n uild n elsti metsurfe nd use to stik to different solid if high onversion is lwys requested in prtil pplitions. The stediness property gretly enhnes the ppliility of the elsti metsurfe in potentil pplitions. n onlusion, we design n ultr-thin elsti metsurfe with ility in mnipulting refleted elsti wve modes. The elsti metsurfe ontins oliquely pled elliptil oting lyers, whih rek the deoupling etween nd T elsti modes nd ring strong non-degenerte dipolr resonne. nput elsti wves n e totlly onverted into the other mode fter refleted y the elsti metsurfe. The totl onversion effet ours with speil frequeny nd inluded ngle. The speil frequeny is the eigenfrequeny of the dipolr resonne, nd the speil inluded ngle depends on ut insensitive to the rtio etween the nd T wve speeds in kground. High onversion ould lwys e otined for different kground mterils, whih rings potentil

10 pplitions in elsti devies. The uthors would like to thnk Professor Z. Y. iu for disussions. This work ws supported y the Ntionl Nturl Siene Foundtion of Chin (Grnts No ). eferene [1] N. Yu, P. Genevet, M. A. Kts, F. Aiet, J.-P. Tetienne, F. Cpsso, nd Z. Gurro, Siene 334, 333 (011). [] Y. i, B. ing, Z.-m. Gu, X.-y. Zou, nd J.-. Cheng, Sientifi eports 3, 546 (013). [3] Y. Xie, W. Wng, H. Chen, A. Konneker, B.-. Pop, nd S. A. Cummer, Nture Communitions 5, 5553 (014). [4] K. Tng, C. Qiu, M. Ke, J. u, Y. Ye, nd Z. iu, Sientifi eports 4, 6517 (014). [5] B. iu, W. Zho, nd Y. Jing, Sientifi eports 6, (016). [6] J. i, C. Shen, A. Díz-uio, S. A. Tretykov, nd S. A. Cummer, Nture Communitions 9, 134 (018). [7]. Ghffrivrdvgh, J. Nikoljzyk,. Glynn Holt, S. Anderson, nd X. Zhng, Nture Communitions 9, 1349 (018). [8] Y. Zhu, J. Hu, X. Fn, J. Yng, B. ing, X. Zhu, nd J. Cheng, Nture Communitions 9, 163 (018). [9] Y. Tin, Q. Wei, Y. Cheng, nd X. iu, Applied Physis etters 110, (017). [10] G. M, M. Yng, S. Xio, Z. Yng, nd P. Sheng, Nture Mterils 13, 873 (014). [11] H. Zhu nd F. Semperlotti, Physil eview etters 117, (016). [1] Y. iu, Z. ing, F. iu, O. Di, A. m, nd J. i, Physil eview etters 119, (017). [13]. Zhu, H. Ysud, G.. Hung, nd J. K. Yng, Sientifi eports 8, 483 (018). [14] H. ee, J. K. ee, H. M. Seung, nd Y. Y. Kim, Journl of the Mehnis nd Physis of Solids 11, 577 (018). [15] S. i, J. Xu, nd J. Tng, Applied Physis etters 11, (018). [16] H. ee, J. H. Oh, H. M. Seung, S. H. Cho, nd Y. Y. Kim, Sientifi eports 6, 406 (016). [17] Y. Xu, Y. i,. Co, Z. Yng, nd X. Zhou, AP Advnes 7, (017). [18]. Co, Z. Yng, nd Y. Xu, Journl of Sound nd Virtion 418, 1 (018). [19] X. Su, Z. u, nd A. N. Norris, Journl of Applied Physis 13, (018). [0] X. Shen, C.-T. Sun, M. V. Brnhrt, nd G. Hung, Journl of Applied Physis 13, (018). [1] J. M. Kweun, H. J. ee, J. H. Oh, H. M. Seung, nd Y. Y. Kim, Physil eview etters 118, (017). []. Zhu, X. N. iu, G. K. Hu, C. T. Sun, nd G.. Hung, Nture Communitions 5, 5510 (014). [3] V.. Gusev, Journl of Applied Physis 117, (015). [4] Z. iu, X. Zhng, Y. Mo, Y. Y. Zhu, Z. Yng, C. T. Chn, nd P. Sheng, Siene 89, 1734 (000). [5]. Zhu, X. N. iu, G.. Hung, H. H. Hung, nd C. T. Sun, Physil eview B 86, (01). [6] P. Pi, A. Shref, Z. Xiujun,. Yn, nd W. Ying, P (urophysis etters) 104, 6001 (013).

11 FG. 1. () Shemti of the unit ell of our periodi system long the x-diretion. Periodi oundry onditions re pplied on the left nd right side, free oundry ondition is pplied on the top side, nd soring oundry ondition is pplied on the ottom side. Plne longitudinl wves re normlly inident from the ottom, nd refleted wves re trnsverse wves. () Shemti of zoom view for one unit ell of the elsti metsurfe. Here r, r nd r re the rdii for steel ore, semimjor nd semiminor xes of the ruer lyer, respetively. p nd re the T period onstnt nd inluded ngle. () Simultion results of the onverting effiieny / s funtion of frequeny nd the inluded ngle. Thik white rrows indite where the totl T onversion effet ( / = 1) ours.

12 FG.. () The nd struture of elsti metmterils onsisting of the sme unit ells s the elsti metsurfe. The inluded ngle is =. Blue dshed lines denote the two eigenfrequenies nd for non-degenerte dipolr resonne. () nd () re the displement field distriutions of eigenmodes on the seond nd third nd t the X points, respetively. Drk red nd drk lue orrespond to 1 nd 0 of the normlized mgnitude, respetively, nd thin white rrows indite diretions.

13 FG. 3. () A shemti view of the mss-spring model. The light red disk nd the greed ring indites the steel with mss m nd the mtrix with mss M, respetively, in one unit ell of the elsti metsurfe. The spring stiffness long the mjor xes for extensionl nd sher displements re nd G, those long the mjor xes re nd G, respetively. The hlf one-dimensionl montomi hin elow M indite the solid kground with mss m nd spring stiffness nd G. () Anlytil results otined from the mss-spring model. The T onverting effiieny / s funtion of frequeny nd the inluded ngle. Thik T white rrows indite where / =1.

14 FG. 4. () The onverting effiieny is s funtion of the ngle with given frequeny or. The red line nd lk dots indite the nlytil results otined from the q. (6) nd the simultion results shown in Figs. 1(), respetively. The green line nd lue squres indite those results with given frequeny. () is the sme with () ut different kground of eryllium. The used mteril prmeters for eryllium re 3 = 1850kg/m, l, = 1800m/s nd t, = 7800m/s.