Interaction of Lamb Waves with Geometric Discontinuities: An Analytical Approach Compared with FEM

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1 Ineacion of Lamb Waves ih Geomeic Disconiniies: An Analyical Appoach Compaed ih FEM Banibaa Podda a) and Vico Gigii a) a) Depamen of Mechanical Engineeing, Univesiy of Soh Caolina, USA Absac. The non-descive esing of maeials can be condced by vaios echniqes. Amongs hese, mehod based on lasonic aves is one of he mos common one. Of hese lasonic aves Lamb aves ae of paicla inees fo he inspecion of lage sces fo vaios easons. Scaeing of Lamb aves fom flas has geneaed a consideable amon of eseach ove las cople of decades. Mos of he ok has been done sing compaional ools like Finie Elemen Mehods and expeimenal echniqe. In his pape an analyical appoach is pesened o develop a fndamenal ndesanding of he scaeing of Lamb aves fom geomeic disconiniies in dimensions. We have consideed simples of all geomeic disconiniy a sep, as his fndamenal ndesanding can easily be exended o coosion o cack. INTRODUCTION The goal of a scal healh monioing sysem is o implemen pocesses o deec and chaaceize damages in engineeing sces. Hee he damage is defined as changes of maeial and/o geomeic popeies of a sce. These sysems may se many diffeen physical phenomena o deec and chaaceize damages. One of hese phenomena is elasic ave popagaion in hin plae like sces, knon as Lamb ave popagaion. To develop a SHM (scal healh monioing) sysem based on Lamb ave popagaion e fis need o ndesand ho Lamb aves ineac ih diffeen damage ypes. In his sdy e aemped o develop an ndesanding of ho Lamb aves ineac ih geomeic disconiniies sing analyical modelling echniqe. Fo simpliciy e consideed only a D sep ype geomeic disconiniy as his addesses he fndamenal challenge of saisfying he coninos bonday condiions a he locaion of he disconiniy iho geing ino a mch complicaed geomeic condiion. Also his case can easily be exended o he case of eal life damages sch as cack o a delaminaion. Nmeos sdies have been done o solve his poblem [ 8]. In mos of hese sdies finie elemen based appoach compaed ih expeimenal esls ee sed o ndesand he scaeed field. In ohe sdies [] modal decomposiion mehod as sed o pedic he scaeed ave field. B he coninos bonday condiion as scaled don o discee poins acoss he bonday o saisfy he bonday condiions as i is vey challenging o saisfy he coninos bonday condiion. In 98, Gegoy e-al [9,] inodced a mehod called poecion mehod o saisfy he coninos bonday condiions. This mehod as also capable of pedicing he singlaiy in sesses in case of a geomeic disconiniy [4]. A modified fom of poecion mehod as sed by Gahn in [6]. Alhogh his scala poecion mehod as simple and able o pedic he scaeed ave field, he convegence as slo [7]. In Moea e-al [7,] sed a veco poecion mehod hich had fase convegence and shoed pomising esls. In his sdy a fom of modal expansion of Lamb modes is sed o calclae he scaeed ave field. To saisfy he coninos bonday condiions e have sed veco poec of he bonday condiions o simplify hem. Since Rayleigh-Lamb eqaion has infinie nmbe of complex oos esling in infinie nmbe of complex Lamb modes, fo he sake of convenience e ill call his mehod complex mode expansion ih veco poecion (CMEP).

2 FIGURE. Axial flexal aves vs. Lamb aves. We ae going o conside only axial-flexal aves a he sep o calclae he scaeed ave field. This ill seve o pposes. Fis, i is easie o ndesand he scaeing pocess sing axial-flexal aves compaed o Lamb aves hich involves complicaed mode shapes. Second, he axial-flexal ave based model ill ac as a benchmak fo he Lamb aves based model as axial-flexal aves ae lo feqency appoximaion of he Lamb aves. This is illsaed in Fig.. INTERACTION OF AXIAL-FLEXURAL WAVES WITH A STEP Le hee be a sep of deph d along he idh in an infiniely long and infiniely ide plae of hickness h a a disance = ih ecceniciy beeen he egion and egion being a as shon in Fig.. Also, le s imagine ha hee is an axial D ave and flexal D ave avelling in +ve diecion in he egion. Le s define hese aves consideing he oigin a = as, i( x) ( x, ) e ˆ i i ( x, ) i e ˆ i( x) i FIGURE. Illsaion of he sep and ineacing D aves. Afe ineacing ih he sep he aves esl in a scae field expessed in ems of all possible modes inclding he evanescen modes. The scae field is expessed as,

3 ( x, ) ( x, ) e ( x, ) e ˆ i[ ( xx) ] e ˆ i[ ( xx) ] ee ˆ ( xx) i ( x, ) ( x, ) e ( x, ) e ˆ i[ ( xx) ] e ˆ i [ ( xx) ] ee ˆ ( xx ) i hee sands fo ansmied ave and sands fo efleced aves. The scaeed aves have nknon complex amplide hich e need o obain by applying bonday condiions. The bonday condiions applied a he sep ae coniniy of slope and displacemens ih foce and momen balance a he neal axis (Fig. ). FIGURE. Bonday condiionss a he sep. This leads s o a se of linea eqaions hich can be expessedd as, AX B i i ia a hee, A EA EA ie I EI ie I EI EI EI iae A EI EI i x e ˆ i ˆ i i x i e ˆ i ˆ i i i x e ˆ ia i B ˆ i x e i ˆ ia ˆ i i i x EA ˆ EA e ˆ i i i x ie I ie I ˆ e i ˆ i i x EI ˆ e E I ˆ i x i hich afe solving fo {X} ih i = and i = e ge,,x ˆ ˆ eˆ ˆ ˆ eˆ and

4 (a) (b) (c) FIGURE 4. (a) Amplide of efleced axial ave, (b) amplide of ansmied axial ave, (c) amplide of efleced flexal ave and (d) amplide of ansmied flexal ave. (d) (a) (b) FIGURE 5. (a) Poe flo in egion and (b) poe flo in egion. The poe flo hogh he sep is expessed as,

Re i i i i E P A Re I E P A Re I Re In Fig. 5 e can see ha he poe flo is balanced acoss he sep. Fom he poe expessions e can noice ha hee is no conibion of he evanescen modes o he poe flo.

5 Re i i i i E P A Re I E P A Re I Re In Fig. 5 e can see ha he poe flo is balanced acoss he sep. Fom he poe expessions e can noice ha hee is no conibion of he evanescen modes o he poe flo. Hoeve i is vey impoan o conside hem fo modal expansion as hey ae solions of he govening eqaion of he ave field. Wiho consideing hose he bonday condiions canno be saisfied. Theefoe e need o find all possible solions of he Rayleigh-Lamb eqaion o apply modal expansion fo saisfying he bonday condiions. SOLVING RAYLEIGH-LAMB EQUATION Since Rayleigh-Lamb eqaion is a anscendenal eqaion and does no have closed fom solion, e have fond he oos of his eqaion ove lage domain of complex plane by sing ecsive ieaive algoihms ien in MATLAM. The algoihm can convege o oo ihin specified accacy ih ceainy. This is impoan as e need o se coec oo of his eqaion fo CMEP. an an 4 d P S d S P S, P S c c P S FIGURE 6. Roos of Rayleigh-Lamb eqaion. Symmeic modes ae ploed in ble and anisymmeic modes ae ploed in ed.

6 INTERACTION OF LAMB WAVES WITH STEP Fo Lamb aves e conside he inciden ave field as, i C e and he scaeed field as, hee, sbscip i sands fo inciden aves and s sands fo scaeed aves ih spescip + fo aves avelling in + diecion and fo aves avelling in diecion. Also hee sbscip epesens diffeen ave modes. The bonday condiions applied a he sep ae displacemen bonday condiion, z b, x and acion bonday condiions,, b z h, x, z, b x, b z h, x, z b, x We poec hem ono an appopiae complee ohogonal veco space o emove he z dependence of he bonday condiion. To ake advanage of he ohogonaliy of sess and displacemen mode shapes of Lamb ave modes, e choose con as he poecion vecos fo he sess bonday condiions and co n bonday condiions independen of z ae, FIGURE 7. Illsaion of he sep and ineacing Lamb aves. s s i C i i C i e s C e s, x i e x i x C s i x s C as he poecion vecos fo he displacemen bonday condiions. The final foms of he e i x e i x

b h dz b dz dz S S S This leads s o he se of linea eqaions expessed as, A C B Fo

consideedd hich shold give s easonably accae esl.

Lamb aves andd e can see ha 7 modes ae moe han enogh fo he esl o convege.

, S i,,, ; ; x x 7 FIGURE 8. Convegence sdy.

he Lamb aves, he scae field fo D aves shold be same as scae field of Lamb aves a lo

7 b h dz b dz dz S S S This leads s o he se of linea eqaions expessed as, A C B Fo nmeical eslss e need o deemine he maximm nmbe of oos of Rayleigh-Lamb eqaion o be consideedd hich shold give s easonably accae esl. Fige 9 shos he convegence of he modal paicipaion facos of he fis hee modes of he Lamb aves andd e can see ha 7 modes ae moe han enogh fo he esl o convege. Then he above eqaion can be easily solved sing maix invesion in MATLAB., S i,,, ; ; x x 7 FIGURE 8. Convegence sdy. COMPARISON WITH D WAVES Since axial and flexal aves ae he lo feqency appoximaion of he Lamb aves, he scae field fo D aves shold be same as scae field of Lamb aves a lo feqency. Fo compaison e have calclaed scaeed field of he D aves and Lamb aves fo sep size of mm in a mm plae ih S mode as he inciden Lamb ave mode. FIGURE 9. Geomey of sep consideed.

8 (a) (b) (c) (d) FIGURE. Compaison of y displacemen a neal plane (a), (c) ansmied aves and (b), (d) efleced aves. FEM MODEL FIGURE. A FEM model. To model ineacion of Lamb aves ih sep e sed D plane sain model. To obain he scae coefficiens as a fncion of feqency e did hamonic analysis. Hamonic analysis nomally ill podce sanding ave field. To ge a ansien esponse e inodced non eflecive bonday a he edges of he model sch ha hey do no eflec any ave. Theefoe doing a hamonic analysis ih noneflecing bonday enabled s o obain he scae coefficiens as a fncion of ime. The noneflecing bonday as ceaed sing linea sping dampe elemen aached o he sface of he bondaies. The damping consans ee vaied sch ha no eflecion occs a he noneflecing bonday. The simlaion as done fo inciden S Lamb ave mode. Fom Fig. e can see ha he esls fom FEM and CMEP mach pefecly. The mino noise in he FEM daa as de o he nmeical noise geneaed by ond-off eos.

9 (a) (b) (c) FIGURE. Compaison of op sface x displacemen (a), (c) ansmied aves and (b), (d) efleced aves. SUMMARY AND CONCLUSION A obs algoihm as developed o find oos of Rayleigh-Lamb eqaion ove a vey complex domain ih conolled accacy. An analyical, CMEP as developed o pedic he scae field of Lamb ave podced by a geomeic disconiniy. Along ih ha a simplified model o pedic scaeing of D aves fom geomeic disconiniy as developed mainly o develop bee ndesanding. Boh he model agee ih each ohe a lo feqency confiming hei validiy. A FEM model as made o compae and validae CMEP. The nmeical model poved CMEP o be an accae pedicion of he scae field. Vice vesa CMEP confimed he validiy of he esl of FEM simlaion. This also poves ha CMEP can be sed o check fo FEM model validiy. CMEP also shos ha i is possible o obain he scae field of Lamb ave analyically. CMEP can also pedic he local field of vibaion in ems of nonpopagaing Lam ave modes. The echniqe of veco poec can be exanded fo D geomeic disconiniies and eal damages sch as cacks and delaminaion. Theefoe CMEP can be exended o model scaeing of Lamb aves in D geomeies iho mch difficlies. ACKNOWLEDGMENTS Sppo fom office of Naval Reseach # N4---7, D. Ignacio Peez, Technical Repesenaive and Ai Foce Office of Scienific Reseach #FA955---, D. David Sagel, Pogam Manage; ae hankflly acknoledged. (d)

10 REFERENCES. D. N. Alleyne and P. Caley, "The ineacion of Lamb aves ih defecs," IEEE Tans Ulason Feoelec Feq Conol, 9 (), 8 97 (99).. D. N. Alleyne and P. Caley, "The effec of disconiniies on he long-ange popagaion of Lamb aves in pipes," Poc Ins Mech Eng, Pa E J Pocess Mech Eng., 7 (996).. M. Casaings, E. Le Clezio, and B. Hosen, "Modal decomposiion mehod fo modeling he ineacion of Lamb aves ih cacks," J Acos Soc Am., (6), (). 4. M. A. Floes-López and Gegoy R. Doglas, "Scaeing of Rayleigh-Lamb aves by a sface beaking cack in an elasic plae," J Acos Soc Am., 9 (4), 4 (6). 5. E. V. Glshkov, N. V. Glshkova, and O. N. Lapina, "Diffacion of Nomal Modes in Composie and Sepped Elasic Wavegides. J Appl Mah Mech., 6 (), 75 8 (998). 6. T. Gahn, "Lamb ave scaeing fom a cicla paly hogh-hickness hole in a plae,". Wave Moion. 7 (), 6 8 (). 7. L. Moea, M. Caleap, A. Velichko, and P. D. Wilcox, "Scaeing of gided aves by fla-boomed caviies ih iegla shapes," Wave Moion, 49 (), (). 8. S. Rokhlin, "Diffacion of Lamb aves by a finie cack in an elasic laye," J Acos Soc Am., 67 (4), (98). 9. Gegoy R. Doglas and I. Gladell, "The canileve beam nde ension, bending o flexe a infiniy," J Elas. (4), 7 4 (98).. Gegoy R. Doglas and I. Gladell, "The eflecion of a symmeic Rayleigh-Lamb ave a he fixed o fee edge of a plae," J Elas.,, 85 6 (98).. L. Moea, M. Caleap, A. Velichko, and P. D. Wilcox, "Scaeing of gided aves by hogh-hickness caviies ih iegla shapes," Wave Moion, 48 (7), ().

Chapter Finite Difference Method for Ordinary Differential Equations

Chapter Finite Difference Method for Ordinary Differential Equations Chape 8.7 Finie Diffeence Mehod fo Odinay Diffeenial Eqaions Afe eading his chape, yo shold be able o. Undesand wha he finie diffeence mehod is and how o se i o solve poblems. Wha is he finie diffeence