A WAVE DISPERSION MODEL FOR HEALTH MONITORING OF PLATES WITH PIEZOELECTRIC COUPLING IN AEROSPACE APPLICATIONS

Transcription

1 4t Middle East NDT Conference and Eibition Kingdom of Barain Dec 007 A WAVE DISPERSION MODEL FOR HEALTH MONITORING OF PLATES WITH PIEZOELECTRIC COUPLING IN AEROSPACE APPLICATIONS Amed Z. El-Garni and Wael G. Abdelraman King Fad University of Petrolem and Minerals Daran Sadi Arabia ABSTRACT Plates wit pieoelectric copling ave been sed etensively for active vibration damping of aerospace strctres. Ensring te eistence of te correct amont of te pieoelectric constitent over time and ealt monitoring are essential in te reliability analysis of tese plates. NDT tecniqes involving wave propagation in solid media proved to be effective inspection tools for aerospace strctres. In tis paper a recently developed continm mitre model for stdying gided wave propagation in bilaminated periodic composites of pieoelectric materials is sed to investigate te effect of pieoelectric copling on te wave dispersion caracteristics of sc plates. Te teory leads to te simple governing copled eqations for te actal composite wic retain te integrity of te propagation process in eac constitent bt allow tem to coeist nder analytically derived interaction parameters. As a conseqence of te analysis effective mitre properties of te composite are obtained in te ero-freqency limit. Te analysis for te cases wit and witot te pieoelectric constitent can be copled wit eperimentally obtained dispersion crves to give an accrate estimate of te percentage of te pieoelectric patc. Te procedre lends itself to modifications tat allow it to be sed as an effective tool in qality control of pieoelectric copled plates manfactring or for ealt monitoring of sc strctres. INTRODUCTION Tere as been an increased interest in determining te effective blk properties of pieoelectric composite materials. See for eample Benveniste 99 Bisegna et al. 996 and Dnn et al. 99. Tis as been peraps prompted by te fact tat since te late seventies pieoelectric composites ave been sed in te manfactring of ig tec components sc as ltrasonic transdcers and actators. In some idealied sitations and for simple systems eact soltions migt be obtained. On te oter and for simple deterministic geometries limited sccess as been realied in calclating some of te properties; tis is based primarily on solving appropriate bondary vale problems. For te most sitations owever properties are calclated or estimated by sing bonds scemes or by sing varios teories of mitre depending pon available information abot te variability in constitent properties geometrical arrangements and interactions. See for eample Nayfe 995 for detailed references. In te case of dynamic applications and in particlar tose involving wave propagation te applicability of te effective modles teories is somewat restrictive. Specially tese teories are incapable of reprodcing te dispersion and prononced alteration spreading and attenation of propagation plses in tese composites. Te necessity to simlate sc effects on te mecanical termal and electromagnetic response of composites as led to development and applications of several teories reflecting te inflence of te microstrctre. Wang 00 derived teoretical

2 4t Middle East NDT Conference and Eibition Kingdom of Barain Dec 007 epressions for te dispersive caracteristics and mode sapes of te transverse displacement and electric potential of te pieoelectric layers and stdied te limits of te wave velocity as te wave nmber increases. Later Qian et al. 004 obtained te dispersion relations for wave propagation in periodic pieoelectric composite layers. In tis paper a continm mitre model tat was developed by Nayfe et al. 999 for te stdy of gided wave propagation in pieoelectric plates is adapted to investigate te effect of te pieoelectric content on te copling caracteristics dispersion properties and te ero-freqency limit of te wave speed. Tis provides a means of estimating te pieoelectric content of composite plates by measring tese caracteristics. Te model employs te se of for copled simple eqations for te propagation process in eac constitent sbject to some interactions. Te derived system of eqations is readily adaptable to te stdy of armonic ecitation in te system. Te tility and range of applicability of tis simple teory was establised by comparison wit an eact treatment. See Nayfe et al FORMULATION OF THE PROBLEM FIELD EQUATIONS Details of te micromecanical model can be fond in Nayfe et al A partial development of te development of tis model is provided ere. Consider te propagation of waves in te direction parallel to te interface of a periodic array of bilaminated composite as sown in figre a. From symmetry all field variables are independent of te y -coordinate. Ts te problem redces to strictly two-dimensional one. For longitdinal wave propagation along te -direction te wave motion is frter restricted to yield symmetric and antisymmetric displacement components. Insring sc symmetries also leads to te vanising of te sear stress and te transverse displacements mecanical and electric at te center of eac layer. Tese symmetry conditions allow te isolation of a repeating nit cell of te composite as sown in figre b. For eac constitent of tis cell a local transverse -coordinate wit te origin located at te layer s center is assigned. Symmetry and te applicable continity conditions are sown on figre c. Wit respect to tis geometrical arrangement te relevant pieoelectric field eqations for eac constitent consist of te momentm eqations + = ρ a t + = ρ t b te carge eqation of electrostatics D D + = 0 te strain-mecanical displacement relations S = S = γ = + te electric field-electric potential relations

3 4t Middle East NDT Conference and Eibition Kingdom of Barain Dec 007 φ φ E = E = 4 and te copled constittive relations E S ij = cijkl S kl ekij Ek Di = eikl S kl + ε ik Ek. 5a b Te elastic pieoelectric and dielectric constants for te ortotropic pieoelectric system are given in te epanded matri forms E E E C C C E E E C C C E E E E C C C C pq = E C E C 55 0 E C66 S e e e 0 0 ε 0 0 = S S e ip e6 ε ij = 0 ε e 0 5 S 0 0 ε wic reflect renaming cijkl as C pq sc tat p and q = 6 are replaced by or or and or respectively. For simplicity of te notation we sall tereafter sppress te sperscripts E and S in te eqations 6 and 8. According to te above relations te constittive eqations 5 take te epanded form φ = C + C + e 9a φ = C + C + e 9b φ = C e 9c φ = e + e ε 9d 5 D φ D = e 5 + ε. 9e wic once again old for eac layer. SYMMETRY AND CONTINUITY CONDITIONS Te above field eqations are spplemented wit te symmetry conditions 0 = 0 0 = 0 D 0 = 0 0a 0 = 0 0 = 0 D 0 = 0 0b tat old at te center of layers and respectively. Te continity conditions at te interface defined by = and = = = a

4 4t Middle East NDT Conference and Eibition Kingdom of Barain Dec 007 = = b = φ D φ D =. c were sperscripts and and sbscripts and refer to layers and respectively. ACROSS-THICKNESS AVERAGING Eliminating te -dependence by performing te across-tickness integration: = 0 d 0 = d. α If te symmetry and continity conditions on α = are taken into accont ten applying te averaging to te eqation of motion a and te carge eqation of electrostatics leads to n = + nρ a t n = + n ρ t. b D n + D = 0 4a D n D = 0. 4b Here = = D = D = D 5 define certain interface interaction terms and n = n = 6 wit = + define te volme fractions for layers and respectively. Similarly te constittive relations 9a and 9d for and D are averaged to yield φ n = nc + C + ne 7a φ n = nc C + ne 7b n D = n e φ + e nε 7c φ nd = ne e nε 7d were = = 8 also defines an interface interaction term. EVALUATION OF THE INTERACTION TERMS APPROXIMATIONS 4

5 4t Middle East NDT Conference and Eibition Kingdom of Barain Dec 007 To solve for te interaction terms we assme tat and D vary linearly wit according to = A = B D = C 9abc and tat and D vary linearly wit according to = A 0a B = 0b =. 0c D C Tese relations are intentionally cosen wic atomatically satisfy te individal symmetry and interface conditions as reqired by eqations 0 and. Sbject to tese approimations we now proceed to average te remaining constittive relations 9c and 9e for eac constitent. Te procedre is smmaried as α follows. First we sbstitte te approimate epressions for and D α α= into eqations 9c and 9e. Second we mltiply te reslting eqations by and for α = and α = respectively. Tird we average in accordance wit eqations ; to finally arrive at = C55 + e5 φ φ a Q = C55 + e5 φ φ b D = e5 ε φ φ c Q D = e5 ε φ φ. d wit n Q =. n INTERFACE TERMS AND By eliminating and interface terms and φ as φ D te for eqations can be solved for te = T + T T4φ T5φ T a = P + P Pφ P4φ b φ were T = e Q + e R + C Q + C T = e Q + e R + C Q 5 5 e5 Q + e e5 Q + e5 T = R + C T = R e Q

6 4t Middle East NDT Conference and Eibition Kingdom of Barain Dec 007 T5 = e5 Q + e5 R5 e5 and T T T4 T5 P = R R P = R R P = R R4 P4 = R R5. T T T T wit e5 Q + e5 e5 Q e5 R = R = R = ε Q + ε ε Q + ε ε Q + ε ε Q ε R 4 = 5 = ε Q + ε ε Q + ε R. INTERACTION TERMS AND D Sbstittion of te epressions for and φ from eqation back into eqations finally reslts in = F + F φ φ 4a D = G + G φ φ 4b were T F = C55 + e5 R C55 + e5 R T T4 F = C55 + e5 R e 5 e5 R4 T T G = e 5 ε R e 5 ε R T T4 G = e 5 ε R ε + ε R4. T INTERACTION TERM Te eqation relating te final interaction term to mst be obtained from te only remaining interface condition namely φ and φ = at =. To tis end sbstitting for teir respective approimations from eqations 9b and 0b and in trn into te epressions 9b for leads to C φ = C + + e 5a C φ = C + e. 5b Invoking te continity relation on finally leads to φ = E E + E E and eliminating 4 φ and φ as per eqation 6 6

7 4t Middle East NDT Conference and Eibition Kingdom of Barain Dec 007 were T T E = C C + e e P E = C C + e e P E T E T T4 T5 E = C C + e e P E 4 = C C + e e P4 E T E T and C C E = +. n n MIXTURE EQUATIONS OF MOTION So far we ave solved for all interaction terms and D in terms of te average field variables φ and φ. Eqations 4 and 7 togeter wit eqations 4 and 6 define te general qasi-one-dimensional model for te pieoelectric system. Tese eqations are now combined into te following system of for copled partial differential eqations φ φ n [ ρ a a a a4 = F + F φ φ ] t 7a n = [ F t 7b φ φ a [ + a a a4 = G + G φ φ ] 7c φ φ a [ 4 + a4 a4 a44 = G + G φ φ ]. 7d Here a = C n + C E a = C E a = e n + C E a4 = C E4 a = C E a = C n + C E a = C E a 4 = e n + C E4 a = e n + e E a = e E a = ε n e E a 4 = e E4 a4 = e E a 4 = e n + e E a 4 = e E a 44 = ε n e E4. Te for copled eqations 7 comprise a mitre system for te laminated pieoelectric composite. In general tese eqations retain te integrity of te propagation process in te individal constitents and allow tem to coeist nder te derived φ φ ρ a a a a 4 + F φ φ interaction parameters and D. In particlar information as to te distribtion of te field variables in te individal constitent is readily obtainable. As a frter observation assming tat te left and sides of eqations 7 remain finite wile letting 0 dictates tat approaces and φ approaces ] 7

8 4t Middle East NDT Conference and Eibition Kingdom of Barain Dec 007 φ. For tis limiting case and φ φ and te sm of eqation 7a and 7b togeter wit te sm of eqation 7c and 7d yield te two copled eqations φ ρ A A = 0 φ c A 0 + A = 8a b t were stands for eiter or. And a similar remark olds for φ. Here A = a + a + a + a A = a + a4 + a + a 4 A = a + a + a4 + a 4 A = a + a4 + a4 + a44 ρ c = ρn + ρ n. e Inspection of eqations 8 sows tat A A = A and A define effective C e e e and ε for te composite respectively. Tese qantities are frter redced and written in terms of te individal constitents properties and volme fractions as nn C C C e = A = nc + nc 9a n C + n C nn C C e e e A = A = ne + ne nc + nc e nn e e = A = nε + nε + nc + nc e = 9b ε 9c ILLUSTRATION As an illstration for te effect of te pieoelectric constitent on te composite dynamic caracteristics we stdy te tree arrangements wit PZT65/5 being te pieoelectric constitents in two of tem and PZT5 in te tird. In all cases we se = 0.5 mm and = 0.5 mm wic reslt in te volme fractions n = 0.5 and n = 0.5. Te material properties sed in te calclations are presented in Table. To qantify te inflence of pieoelectric copling Figres -4 depict te e e e variation of te effective properties C e and ε wit te PZT65/5 volme fraction. Tese figres sow caracteristics similar to tose predicted by Ald et al. 99. By correctly selecting te freqency of te propagating wave and measring te corresponding vale of effective properties tese figre provide a means for predicting te pieoelectric content in te composite. If applied over longer periods of time it can provide sefl information regarding te deterioration of te material properties wit time. CONCLUSION In tis work a previosly developed model for te wave propagation in pieoelectric composite plates is tilied to stdy te effective properties of several pieoelectric arrangements. As a conseqence of te analysis te variations of te effective stiffness pieoelectric copling and permittivity wit te pieoelectric volme fraction of te composite are obtained. Te work can be sed to estimate te pieoelectric volme fraction based on measrements of tese effective properties. 8

9 4t Middle East NDT Conference and Eibition Kingdom of Barain Dec 007 ACKNOWLEDGMENTS Te ators are gratefl to King Fad University of Petrolem and Minerals for its spport to tis work. REFERENCES. Benveniste Y. 99 Universal Relations in Pieoelectric Compositeswit Eigenstress and Polariation Fields II: Mltipase Media Jornal of Applied Mecanics Volme 60 pages Bisegna P. and Lciano R. 996 Variational Bonds for te Overall Properties of Pieoelectric Composite Jornal of Mecanics and Pysics of Solids Volme 44 Nmber 4 pages Dnn M.L. and Taya M. 99 Micromecanics Predictions of te Effective Electroelastic Modli of Pieoelectric Composites International Jornal of Solids and Strctres Volme 0 pages Nayfe A. H. 995 Wave propogation in layered anisotropic media wit application to composites Amsterdam Elsevier. 5. Nayfe A. H. Faidi W. Abdelraman W. G. 999 An Approimate Model for Wave Propagation in Pieoelectric Materials I-Laminated Composites Jornal of Applied Pysics Volme 85 Nmber 4 pages Qian Z Jin F Wang Z and Kisimoto K 004 Dispersion Relations for SH-Wave Propagation in Periodic Pieoelectric Composite Layered Strctres International Jornal of Engineering Science Volme 4 Nmber 7 pages Smit W. A. and Ald B. A. 99 Modeling - Composite Pieoelectrics: Tickness-mode Oscillations IEEE Transactions Ultrasonic Ferroelectrics and Freqency Control Volme 8 pages Wang Q 00 SH Wave Propagation in Pieoelectric Copled Plates IEEE Transactions Ultrasonic Ferroelectrics and Freqency Control Volme 49 Nmber 5 pages

10 4t Middle East NDT Conference and Eibition Kingdom of Barain Dec 007 Figre : a Te geometry of te composite model. Layer Layer = 0 = - = = 0 Figre : b Unit cell geometry and coordinate system. = 0 = 0 D = 0 Layer = = D = = = D φ = φ Layer = 0 = 0 D = 0 Figre : c Unit cell symmetry and continity conditions. 0

11 4t Middle East NDT Conference and Eibition Kingdom of Barain Dec 007 e 0 0 N/m C PZT65/5-Sprr PZT65/5-qart PZT5-Sprr PZT Volme Fraction % Figre. Variation of effective stiffness wit te pieoelectric volme fraction.

12 4t Middle East NDT Conference and Eibition Kingdom of Barain Dec 007 Colmb/m e e PZT65/5-Sprr PZT65/5-qart PZT5-Sprr PZT Volme Fraction % Figre. Variation of effective pieoelectric copling wit te pieoelectric volme fraction.

13 4t Middle East NDT Conference and Eibition Kingdom of Barain Dec PZT65/5-Sprr PZT65/5-qart PZT5-Sprr 0-9 Farad/m ε e PZT Volme Fraction % Figre 4. Variation of effective permittivity wit te pieoelectric volme fraction.

14 4t Middle East NDT Conference and Eibition Kingdom of Barain Dec 007 Table I. Te Material Properties of Selected Materials. Units of C pq and e ip are 0 9 N/m and colomb/m respectively. ε ij is given non- s S S dimensional as ε / ε o were ε o = =0-9 /6π farad/m. Materials Constants PZT 65/5 PZT5 Sprr Qart ρg/cm C C C C C C C 44 C 55 C e e e e 6 e 5 ε ε ε

15 4t Middle East NDT Conference and Eibition Kingdom of Barain Dec 007 Figre a. Composite model geometry. Figre b. Unit cell geometry and coordinate system. Figre c. Symmetry and continity conditions. Figre. Variation of effective stiffness wit te pieoelectric volme fraction. Figre. Variation of effective pieoelectric copling wit te pieoelectric volme fraction. Figre 4. Variation of effective permittivity wit te pieoelectric volme fraction. 5