843. Efficient modeling and simulations of Lamb wave propagation in thin plates by using a new spectral plate element

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1 843. Efficint modling and simulations of Lamb wav propagation in thin plats by using a nw spctral plat lmnt Chunling Xu, Xinwi Wang Stat Ky Laboratory of Mchanics and Control of Mchanical Structurs aning Univrsity of Aronautics and Astronautics, o. 9 Yudao Strt, aning 6, China ylfxcl@6.com, wangx@nuaa.du.cn (Rcivd April ; accptd 4 Sptmbr ) Abstract. A nw spctral plat lmnt is prsntd for modling slctivly or simultanously th symmtric and/or anti-symmtric mods of Lamb wavs propagating in thin plat structurs. Th Lgndr polynomials and th xtndd form of th displacmnt fild ar usd in th formulation. Th diagonal mass matrix is obtaind by using a simpl mthod with lss computational ffort. Dtaild drivations ar providd. Comparisons with xisting rsults ar prformd to validat th formulations as wll as th writtn programs. umrical calculations hav bn carrid out for thin aluminum plats with and without damags by using th proposd spctral plat lmnt. Comparisons rval that th proposd spctral plat lmnt is mor ffctiv than th spctral plat lmnt basd on Chbyshv polynomials and th 3-D spctral finit lmnts with th sam ordr of Lgndr polynomials as th shap functions. Kywords: spctral plat lmnt, wav propagation, Lamb wav, damag idntification.. Introduction It is wll known that dtction of small damag in structurs is an important but a challnging task in nginring practic. Thus, it is a hot rsarch topic in th ara of th structural halth monitoring (SHM). To dvlop ffctiv Lamb wav basd SHM systms it is important to undrstand thoroughly th bhavior of th Lamb wav propagation in structurs with and without damags ithr xprimntally or numrically. It is known, howvr, that th convntional finit lmnt mthod (FEM) is computationally infficint in analyzing lastic wav propagations []. Thrfor, rsarch fforts hav bn focusd on dvloping fficint numrical algorithms for solutions of lastic Lamb propagations for th past fw dcads and ar still undrway. Various numrical algorithms rportd in th litratur, to list only a fw, includ th boundary lmnt mthod (BEM) [], th mass-spring lattic modls (MSLM) [3], th local intraction simulation approach (LISA) [4], th discrt singular convolution (DSC) algorithm [5], and th spctral finit lmnt mthod (SFEM) [6-]. For th spctral finit lmnt (SFE) mthods, thr ar two diffrnt kinds of mthods availabl, namly, th fast Fourir transform basd SFE and th orthogonal-polynomial-basd SFE. Rsarch works show that th SFE mthod basd on th orthogonal polynomials is much mor suitabl for analyzing wav propagation in structurs with complx gomtry than th FFT-bass mthod. Th orthogonal-polynomial-basd SFE mthod is similar to th classical finit lmnt mthod in th assmblag of structural mass and stiffnss matrics, as wll as th solution procdurs. If th shap functions ar Lgndr polynomial, th mass matrix is approximatly in a diagonal form, a rmarkabl advantag ovr th convntional finit lmnt mthod []. Thrfor, th orthogonal- polynomial-basd SFE mthod has bn widly usd to simulat wav propagation in structurs for damag dtction [-5]. Kudla t al. [] studid th wav propagation in -D structurs by using th Lgndr-polynomial-basd SFE mthod. Zak t al. [] invstigatd wav propagation in plats with a crack by using -D spctral finit lmnt and idntifid th crack through th transmittd and rflctd wavs. Png t al. [3] invstigatd th wav propagation in plats by using a 3-D spctral lmnt. Basd on 87

2 Mindlin s thory, Kudla t al. [4] studid th wav propagation in composit plats. Rcntly, Zak [5] proposd an xtndd form of th displacmnt fild and prsntd a -D nw spctral plat lmnt. Th attractiv fatur of th -D plat lmnt is that it can modl slctivly or simultanously th symmtric and/or anti-symmtric mods of Lamb wavs propagating in plat structurs. In othr words, th -D plat lmnt has th capability of a 3-D spctral lmnt [3] in modling th wav propagations in plats. Compard to th 3-D spctral lmnt, th -D plat lmnt is simplr and mor computationally fficint. It is notd that, howvr, th mass matrix of th novl plat lmnt in [5] is not in a diagonal form. Thrfor, additional ffort has to b mad in ordr to allow crucial rduction of th complxity and th cost of th tim intgration by using th cntral finit diffrnc mthod. To rmov this dficincy and rais th computational fficincy furthr, a modifid spctral plat lmnt is proposd hrin. Instad of th Chbyshv polynomials usd in [5], th Lgndr polynomials ar usd as th shap functions and a simpl mthod is usd to formulat th diagonal mass matrix. Ths modifications will rais th critical tim stp (th largst tim stp for stabl tim intgration) mor than 8 % ovr that of th original plat lmnt. Formulations and solution procdurs ar workd out in dtail. To validat th formulations, th writtn programs, as wll as th computational fficincy, svral xampls ar analyzd by using th proposd plat lmnt. umrical rsults ar compard with xisting (rcalculatd) data by using th -D spctral plat lmnt in [5] and 3-D spctral lmnt in [3]. Basd on th rsults rportd hrin, som conclusions ar drawn.. Formulations of th nw spctral plat lmnt. Dfinition of lmnt nods It is known that in th formulation of a tim-basd spctral finit lmnt, th nodal coordinats of an lmnt ar a ky factor that strongly influncs th lmnt prformanc [5]. Diffrnt from th convntional finit lmnt, nods ar distributd non-uniformly in th lmnt. For th fficincy considrations, th nodal coordinats of th lmnt in th lmnt local coordinat systm ξη ar dfind as th roots of th following polynomials []: ξ ξ = η η = () ( ) P ( ) ; ( ) P ( ) whr ξ, η [,], P ( ξ ) and P ( η) dnot th first drivativ of Lgndr polynomials of dgr. If th fifth ordr polynomials ( = 5 ) ar chosn, th nodal coordinats of th lmnt can b spcifid by: ( η ) ( η ) ( η ) ( η ) ( η ) ( η ) ξ or = ξ or = ξ3 or = ( ξi, η ), i, =,...,6 ξ4 or 4 = ξ5 or 5 = ξ6 or 6 = () Eq. () dfins th Gauss Lobatto Lgndr (GLL) points []. Thus, a 36-nod -D spctral plat lmnt can b formulatd. It should b pointd out that th nodal coordinats 88

3 adoptd in th prsnt formulation diffr from th ons usd by Zak [5], thus th spctral plat lmnt is rgardd as a nw on, although th sam xtndd form of th displacmnt fild of th plat is usd in th formulation.. Shap functions Similar to th convntional finit lmnt mthod, th gnralizd displacmnt fild in th 36-nod -D spctral plat lmnt can b assumd as: ξ η = i ξ η ξ i η = i ξ η ξi η i= = i= = (3) q (, ) ( ) ( ) q (, ) (, ) q (, ) whr ( ξ, η) ( ξ ) ( η) = ar shap functions, q (, ) ( i,,,...,6) i i ξ η = dnot th nodal i dgrs of frdom, and i ( ξ ) and ( η ) ar th on-dimnsional shap functions in th local ξη coordinat systm dfind as: ξ ξ η η ( ξ ) = ; ( ) =, i, =,...,6 i 6 6 k k η k=, k iξi ξk k=, k η ηk (4). 3 Displacmnt and strain filds To us a -D spctral lmnt to modl th 3-D bhavior of th Lamb wav propagation in plats, th xtndd form of th displacmnt fild, proposd by Zak [5], is adoptd. Th 3-D displacmnt fild of a plat in th global coordinat systm (x-y-z) st in th middl plan of th plat can b writtn as [5]: u( x, y, z) = u ( x, y) + z Φ( x, y) v( x, y, z) = v ( x, y) + z Ψ( x, y) w( x, y, z) = w ( x, y) + z Ω( x, y) (5) whr u, v and w ar th avrags of th displacmnts at th uppr and lowr surfacs of th plat, Φ, Ψ and Ω ar th diffrncs btwn th displacmnts at th uppr and lowr surfacs, u( x, y, h / ) + u( x, y, h / ) u( x, y, h / ) u( x, y, h / ) u ( x, y) =, Φ ( x, y) = h v( x, y, h / ) + v( x, y, h / ) v( x, y, h / ) v( x, y, h / ) v ( x, y) =, Ψ ( x, y) = h w( x, y, h / ) + w( x, y, h / ) w( x, y, h / ) w( x, y, h / ) w ( x, y) =, Ω ( x, y) = h (6) whr h is th plat thicknss. It is sn that thr ar totally six gnralizd displacmnt functions, thrfor, ach nod of th lmnt has six DOFs, namly, u ( ξi, η ), v ( ξi, η ), w ( ξ, η ), Φ( ξ, η ), Ψ( ξ, η ), Ω ( ξ, η ). i i i i For small strains, th 3-D strain-displacmnt rlations of a plat can b writtn by: 89

4 u Φ w Ω ε x = + z, γ yz = + z +Ψ x x y y v Ψ w Ω ε y = + z, γ zx = + z +Φ y y x x ε, u Φ v Ψ z y y x x z =Ω γ xy = + z + + (7). 4 Mass and stiffnss matrics For a 36-nod -D spctral plat lmnt, th displacmnt fild within th plat lmnt can b xprssd in th following form: u ( ξi, η ) v ( ξi, η ) u 6 6 w ( ξi, η ) v = [ ] { q} = [ i ( ξ, η)] (8) i= = ( ξi, η ) w Φ Ψ( ξi, η ) Ω( ξi, η ) whr [ ] dnots th matrix of shap functions. Th strain within th lmnt can b xprssd as: ε (, ) x u ξi η v ( ξi, η ) ε y ε 6 6 w ( ξi, η ) z = [ B] { q} = [ Bi ( ξ, η, z)] ( B( ξ, η) z B( ξ, η) ){ q} γ = yz i= = ( ξi, η ) + (9) Φ γ zx Ψ( ξi, η ) γ xy Ω( ξi, η ) whr [ ] B is th strain matrix, and { } q is th nodal displacmnt vctor. Instad of using th xisting GLL quadratur rul [], th diagonal mass matrix is computd by using Gaussian quadratur as: 3 3 (, ) (, ) (, ) m = ξ η µ da= µ ξ η J ξ η dξdη II A kp L L kp L i kp i i i= = ( ) ( ) µ H H ( ξ, η ) J ξ, η k, p=,,...,6, I = 6 k - + p () whr µ = ρ ( =,,3) and µ ρ ( ) 9 L h L 3 L = h / L= 4,5,6, ρ is th mass dnsity of th matrial, J ( ξ, η ) is th dtrminant of th Jacobian matrix, H, H and ξ, η ar wights and abscissas of Gaussian quadratur. ot that th computational ffort in obtaining th mass matrix is only about 5 % of that by th xisting GLL quadratur rul. Th stiffnss matrix can b computd by using Gaussian quadratur as: i i

5 T ( ( ξ η) ( ξ η) ) ( ξ η) ( ξ η) V i= = ( ) [ K] = B, + z B, [ D] B, + z B, dv 3 T h T = h [ B] [ D][ B] J( ξ, η) dξdη [ B] [ D][ B] J( ξ, η) dξ dη + ( ) T = h H H [ B ( ξ, η )] [ D][ B ( ξ, η )] J ξ, η + 3 h i= = i i i i i i T H H [ B ( ξ, η )] [ D][ B ( ξ, η )] J ξ, η i i i i ( ) i i () whr [ D ] is th lasticity matrix of th matrial. For an isotropic matrial, [ D ] is givn by: ν ν ν ν ν ν E( ν ) [ D] = 5( ν ) ( + ν) ( ν ) ( ν ) 5( ν ) symmtric ( ν ) ν ( ν ) () whr E is th modulus of lasticity, ν is Poisson s ratio. A corrction factor [6] has bn includd for calculations of th transvrs shar strsss.. 4 Th tim intgration schm In trms of th spctral plat lmnt, th wll-known govrning diffrntial quations for th wav propagation in plat structurs can b writtn in th following matrix form, { } + { } = { } [ M ] Qɺɺ [ K ] Q F (3) whr { Q } is th gnralizd displacmnt vctor of th structur and th doubl dots dnot th scond-ordr drivativ with rspct to tim t, [M] and [K] ar th structural mass matrix and stiffnss matrix, and { F } is th vctor of th tim dpndnt xcitation signal, rspctivly. Du to th fact that th mass matrix [M] is a diagonal matrix, a crucial rduction of th complxity and th cost of th numrical tim intgration can b achivd by using cntral finit diffrnc mthod [7-8]. Thus, Eq. (3) can b xplicitly intgratd as follows, namly, ( ) { } { } { } [ ] { } [ ]{ } Q = t M F K Q + Q Q (4) t+ t t t t t t whr t is th tim, and t dnots th tim stp of th tim intgration. Whn t t, th cntral finit diffrnc mthod is stabl. t = / ω, whr ω max is th maximum circular cr max cr 9

6 frquncy of [ k] ω [ M] =. [ ] and [ ] smallst lmnt in th spctral finit lmnt modl. 3. umrical xampls and discussions k M ar th stiffnss and mass matrics of th Svral xampls with xisting rsults ar invstigatd to validat th formulations, solution procdurs and writtn programs. Zro initial conditions ar assumd for all numrical simulations prsntd in this papr. It should b pointd out that all xisting rsults usd for comparisons ar r-producd by th prsnt authors. 3. Comparisons with SFE basd on Chbyshv polynomials To compar th nw spctral finit lmnt (SFE) basd on th Lgndr polynomials with th sam kind SFE but basd on Chbyshv polynomials (SFE-Chbyshv) [5], considr first th lastic wav propagation in an aluminum thin plat with all four dgs fr and a gomtric configuration of mm mm mm shown in Fig.. Thr spcial points, dnotd by A, E and O, will b usd in th discussions. For vrifications, matrial proprtis and xcitation signal ar th sam as thos in [5], namly, Young s modulus E = 7.7 GPa, mass 3 dnsity ρ = 7 kg/m, and Poisson s ratio ν =.33, and th xcitation signal shown in Fig. c is: F( t) = 5sin( πt /)* ( cos( πt / 6 )) if t 6 and if t> 6 (5) whr th unit of tim t is in micro-sconds ( µs ). Two loading typs shown in Fig. a and b ar considrd. Fig.. Gomtry of an aluminum plat A uniform msh of 5 5 spctral plat lmnts is usd in modling th aluminum squar plat, rsulting in 378,6 dgrs of frdom, th sam as that in [5]. Th total tim is st to.4 ms. For th lmnt with th sam siz and th sam numbr of nods, th critical tim stp is diffrnt. =.88 µs for SFE and =.3 µs for SFE-Chbyshv. Thus, much t cr t cr 9

7 largr tim incrmnt can b usd as compard to SFE-Chbyshv to achiv th sam solution accuracy. In turn, th computational fficincy is raisd by using th proposd lmnt. For loading cas shown in Fig. a, th anti-symmtric Lamb mod A is xcitd. In th prsnt -D plat modl, th dual forcs ar applid at point O along th dirction of w. Fig. 3 shows th rspons of th displacmnt of th plat at th xcitation point O and th boundary point A, and rsults obtaind by both lmnts ar clos to ach othr. Fig.. Loading typs and xcitation signal in tim domain (a) (b) Fig. 3. Rspons displacmnt signals at (a) point O and (b) point A for th A mod For loading cas shown in Fig. b, dual forcs with opposit dirctions ar applid on both th uppr and th lowr surfacs of th plat at point O, thus th symmtric Lamb mod S is xcitd. In th -D plat modl, th dual forcs ar applid at point O along th dirction of th gnralizd displacmnt Ω. Fig. 4 shows th rspons of th displacmnt of th plat at th xcitation point O and th boundary point A. It is obsrvd that rsults obtaind by both lmnts ar again clos to ach othr. Fig. 5 indicats two snap shots at tim of.8 ms (Fig. 5a) and.6 ms (Fig. 5b) and dmonstrat th procss of wav propagation visually. Th rflctiv wavs from th boundaris can b clarly sn in Fig. 5b. 93

8 (a) (b) Fig. 4. Rspons displacmnt signals at (a) point O and (b) point A for th S mod 94 (a) (b) Fig. 5. Propagation of th S mod of Lamb wavs in an aluminum plat 3. Comparisons with th 3-D SFE To compar th nw spctral plat lmnt with th 3-D SFE basd on th sam Lgndr polynomials in [3], considr again th lastic wav propagation in an aluminum thin plat shown in Fig.. Howvr, th thicknss changs to mm and Young s modulus E = 7. GPa. Two loading typs shown in Fig. 6a and 6b ar considrd. Th xcitation signal shown in Fig. 6c is givn by: ( ) ( π ) ( ( π )) F t = sin t / 5 * cos t / 5 / for t 5 and for 5 < t (6) whr th unit of tim t is in micro-sconds ( µs ). For comparisons, th aluminum plat undr invstigation is mshd by. For th nw spctral finit lmnts, thr ar total of,56,6 dgrs of frdom. For th 3-D SFE [3], thr points ar usd in th thicknss dirction ( ) thus thr ar a total of,59,9 DOFs. For th sam siz lmnt, th critical tim stp is diffrnt. t cr =.7 µs for SFE ( 6 6 ) and t cr =.98 µs for 3-D SFE. Th total calculation tim is st to.3 ms. Thus, SFE nds 3 tim stps and 3-D SFE uss 4 tim stps [3] to achiv similar accuracy, sinc th thicknss of th plat is only mm.

9 (c) Fig. 6. Th loading typs and xcitation signal in tim domain For loading cas 3 shown in Fig. 6a, both symmtric mod S and anti-symmtric mod A ar xcitd. In th -D plat modl, th forc is applid at point O along th dirction of th displacmnt componnts v and Ψ. Fig. 7a shows th rspons of th displacmnt v of th plat rcordd at point E. It is obsrvd that rsults obtaind by th two mthods ar comparabl but hav slight diffrncs at crtain tim instants. Prhaps this might b causd by th slightly diffrnt distributions of th in-plan displacmnts along th thicknss dirction. (a) (b) Fig. 7. Rspons signals of displacmnt v at point E For loading cas 4 shown in Fig. 6b, only th symmtric Lamb mod S is xcitd. In th -D plat modl, th dual forcs ar applid at point O along th y dirction. Fig. 7b shows th rspons of th displacmnt v of th plat rcordd at point E. It is obsrvd that rsults obtaind by th two mthods ar clos to ach othr Damag dtction by using th SFE mthod In this sction, a plat with damags shown in Fig. 8 is considrd. Th dimnsion and matrial paramtrs ar th sam as thos givn in Sction 3.. Thr ar two damag zons: d and d. Th damag zon d has th dimnsions of lngth 4 mm and width mm and th damag zon d has th dimnsions of lngth mm and width mm. Load cas is usd in th simulation. Th plat is modld by 36-nod SFEs. Th damag is modld by simply rducing th lmnt stiffnss [9]. In th simulation, modulus of lasticity in th damagd ara is rducd to % of th modulus of lasticity in th undamagd ara. Th total calculation tim is st to.8 ms so th rflctd wavs from th damag can rach all masuring 7 points (points B, C, D and E). Th tim incrmnt is st to s. 95

10 Considr first only th damag d xisting in th plat. Fig. 9 shows a snapshot at tim instant.6 ms. Th rflctd wav from th damag can b clarly obsrvd. For damag dtction, th displacmnt rsponss at points B, C, D and E dpictd in Fig. 8 ar rcordd. Th rsponss of undamagd plat ar thn subtractd from th corrsponding rsponss of th damagd plat. Th diffrncs of rsponss ar plottd in Fig.. Basd on th tim intrval, th wav propagation spd, as wll as th location of points B, C, D and E, th location of th damag can b dtrmind by using th mthod prsntd in []. Fig. 8. A squar plat with damags Fig. 9. Snapshot at tim of.6 ms (a) (b) (c) (d) Fig.. Diffrnc rspons signals of plat with damag d at point (a) B, (b) C, (c) D, (d) E 96

11 Considr nxt two damags xist in th plat. Fig. shows thr snapshots at diffrnt tim instancs to dmonstrat th procsss of th wav propagation. Th rflctd wavs from th both damags ar clarly obsrvd. Similarly, th displacmnt rsponss at points B, C, D and E ar rcordd for damag dtction; th rsponss of undamagd plat ar thn subtractd from th corrsponding rsponss of th damagd plat which ar shown in Fig.. Basd on th tim intrval, th wav propagation spd, as wll as th location of points B, C, D and E, th location of th damags can b dtrmind by using th mthod prsntd in []. Comparison of rsults providd in Fig. and Fig. indicats obvious diffrnc du to diffrnt damags xisting in th plats. (a) (b) (c) Fig.. Propagation of th A mod Lamb wavs in an aluminum plat with damags (a) (b) (c) (d) Fig.. Diffrnc rspons signals of plat with damags at point (a) B, (b) C, (c) D, (d) E 97

12 4. Conclusions In this papr, a nw spctral plat lmnt is proposd for modling slctivly or simultanously th symmtric and/or anti-symmtric mods of Lamb wavs propagating in plats. Th xtndd form of th displacmnt fild and th Lgndr polynomials ar adoptd in th formulation. A simpl way to formulat th diagonal mass matrix is usd to rduc furthr th computational ffort. Formulations ar givn in dtail. Various numrical simulations hav bn carrid out for aluminum plats with and without damag by using th proposd spctral plat lmnt. Comparisons of simulatd rsults to xisting rsults vrifid th formulations. Th fatur of th proposd lmnt, i.., modling slctivly symmtric and/or anti-symmtric mods of Lamb wavs propagating in plats, has bn dmonstratd. It is dmonstratd that th proposd spctral plat lmnt is mor ffctiv than th spctral plat lmnt basd on Chbyshv polynomials and th 3-D spctral finit lmnt basd on th sam ordr of Lgndr polynomials. Sinc th damag location can b also idntifid from th simulatd transmittd and rflctd wavs by using th mthod prsntd in [], thrfor, th nw spctral plat lmnt may b usful in th ara of Lamb-wav basd structural halth monitoring. Acknowldgmnts Th work is partially supportd by th ational atural Scinc Foundation of China (583) and by th Priority Acadmic Program Dvlopmnt of Jiangsu Highr Education Institutions. Rfrncs [] Kim Y., Ha S., Chang F. K. Tim-domain spctral lmnt mthod for built-in pizolctricactuator-inducd Lamb wav propagation analysis. AIAA Journal, Vol. 46, Issu 3, 8, p [] Cho Y., Ros J. L. A boundary lmnt solution for a mod convrsion study on th dg rflction of Lamb wavs. Journal of Acoustical Socity of Amrica, Vol. 99, Issu 4, 996, p [3] Dlsanto P. P., Mignogna R. B. A spring modl for th simulation of th ultrasonic pulss through imprfct contact intrfacs. Journal of Acoustical Socity of Amrica, Vol. 4, 998, p. 8. [4] Dlsanto P. P., Whitcomb T., Chasklis H. H., t al. Connction machin simulation of ultrasonic wav propagation in matrials. I: Th on-dimnsional cas. Wav Motion, Vol. 6, 99, p [5] Wang X., Xu C. L., Xu S. M. Th discrt singular convolution for analyss of lastic wav propagations in on-dimnsional structurs. Applid Mathmatical Modlling, Vol. 34,, p [6] Komatitsch D., Barns C. H., Tromp J. Simulation of anisotropic wav propagation basd upon a spctral lmnt mthod. Gophysics, Vol. 4,, p [7] Igawa H., Komatsu K., Yamaguchi I., t al. Wav propagation analysis of fram structurs using th spctral lmnt mthod. Journal of Sound and Vibration, Vol. 77, 4, p [8] Palacz M., Krawczuk M., Ostachowicz W. Th spctral finit lmnt modl for analysis of flxural-shar coupld wav propagation. Part : Laminatd multilayr composit. Composit Structurs, Vol. 68, 5, p [9] Sun H., Zhou L. Analysis of damag charactristics for crackd composit structurs using spctral lmnt mthod. Journal of Vibronginring, Vol. 4, Issu,, p [] Sriani G. 3-D larg scal wav propagation modling by spctral lmnt mthod on Cray T3E multiprocssor. Computr Mthods in Applid Mchanics and Enginring, Vol. 64, Issu -, 998, p [] Kudla P., Krawczuk M., Ostachowicz W. Wav propagation modling in D structurs using spctral finit lmnts. Journal of Sound and Vibration, Vol. 3, 7, p

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