8th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynaics, and Materials Conference<br>5th 3-6 April 7, Honolulu, Hawaii AIAA 7-75 Modeling of Guided-wave Excitation by Finite-diensional Piezoelectric Transducers in Coposite Plates Ajay Raghavan and Carlos E.S. Cesnik Departent of Aerospace Engineering, University of Michigan, 3 Beal Ave., Ann Arbor, Michigan 89- This paper addresses odeling of guided-wave excitation by surface-bonded piezoelectric wafer transducers in ulti-layered coposite plate structures. Each of the individual layers is assued to have unidirectional fibers in an epoxy atrix and is odeled as being transversely isotropic. The piezoelectric actuators are odeled as causing shear traction along their edges on the plate surface and uncoupled actuator-substrate dynaics are assued. The forulation is generic enough to accoodate arbitrary shape actuators, and specific expressions are derived for the cases of rectangular and ring-shapes. The surface and interfacial conditions are ipleented using the global atrix approach. Since the three-diensional guided-wave field is odeled without using reduced structural forulations, all possible guided-wave odes are captured. The two-diensional spatial Fourier transfor is used to solve the proble, and a rigorous inversion procedure is outlined. These odels are ipleented nuerically and results in the for of haronic radiation plots are presented for various configurations. I. Introduction he prospect of having aerospace and other structures instruented with on-board daage prognosis systes has T generated a lot of interest in the area of structural health onitoring (SHM over the past decade. The hope is that such systes would be able to regularly scan the structure for daage and warn the user in near-real tie about any incipient daage. They should also be able to tie in with prognostic algoriths to furnish estiates about the reaining service life of the structure. The presence of daage prognosis systes ay increase safety. In addition, they could enable a transition fro schedule-driven inspection to condition-based aintenance. The onetary and labor savings benefits of such systes ay be also very significant. Another growing trend in aerospace structures is the widespread use of coposites. The priary advantage of using coposites is their higher stiffness-to-ass ratio copared to etals, which translates into significant fuel and operational-cost savings. In addition, they have better corrosion resistance and can be tailored for preferentially bearing loads along specific directions. However, they are ore susceptible to ipact daage. Ipact can cause daage in the for of delainations or cracks, reduce load-bearing capability, and potentially lead to structural failure. The capability of health onitoring could increase confidence in the use of coposite structures by alerting operators about daage fro unexpected ipact events. A. Guided-wave structural health onitoring While several approaches have been exained for SHM, guided-wave (GW ethods have shown potential to actively interrogate large structural areas with a sparse network of transducers. These essentially involve exciting the structure with high frequency stress waves and processing the difference in structural response with respect to a baseline signal for the pristine condition, fro which daage, if present, can be detected and characterized. A detailed survey on GW SHM, including fundaentals and early history, is presented in a review paper by the authors. GWs can be defined as stress waves forced to follow a path defined by the aterial boundaries of the structure. Aerospace vehicles usually consist of different substructures, each of which can act as waveguides, thereby aking the attractive application areas for GW SHM. In SHM, typically surface-bonded piezoelectric wafer transducers (called piezos in this work are used. Graduate Research Assistant and AIAA Meber Associate Professor and Associate Fellow, AIAA; Corresponding Author; Eail: cesnik@uich.edu; Phone: - 73-76-3397; Fax: -73-763-578 Aerican Institute of Aeronautics and Astronautics Copyright 7 by Ajay Raghavan and Carlos E.S. Cesnik. Published by the Aerican Institute of Aeronautics and Astronautics, Inc., with perission.
However, GW SHM has soe caveats associated with it. In isotropic structures, typically, ore than one ode is possible in a waveguide at any frequency. Furtherore, as shown in Figure (a, each ode has a unique dispersion curve, which represents the relation between phase velocity and frequency. In coposite structures, this is further coplicated by the directional dependence of wavespeeds, due to the difference in elastic properties along different directions (e.g., see Figure (b. Therefore, it is iportant to have a fundaental understanding of GW propagation and characterize the GW field excited and sensed by the transducers being used. Earlier work by the authors, 3 has addressed this issue for isotropic structures. With coposite aterials becoing increasingly coon in aerospace structures, there is a need to address that in coposite plate structures as well. That is the ai of the present paper. B. Previous work The theory of free GW propagation in isotropic, anisotropic and layered aterials for various geoetries as well as excitation using conventional NDE ultrasonic transducers is well-docuented. The free GW odes in isotropic plates and shells were first studied by Lab 5 and Gazis 6 respectively using the theory of elasticity. Earlier works on odeling excitation of GW fields using the theory of elasticity ostly used two-diensional (-D odels, wherein variations along one direction in the plane of the plate were ignored. Work done on odeling excitation in isotropic structures has been reviewed in the authors survey paper. Soe works have exained GW excitation by transducers for non-destructive testing (NDT in coposites. Ditri and Rose 7 used -D elasticity odels along with the noral odes expansion technique to describe GW excitation in coposites. Mal 8 and Lih and Mal 9 developed a theoretical forulation to solve for the proble of forced GW excitation by finite-diensional sources using 3-D elasticity in ultilayered coposite plates. The Fourier spatial integrals were inverted using a nuerical schee. Viscoelastic daping was addressed, and specifically, the cases of excitation by NDT transducers and acoustic eission were solved based on the developed forulation. More recently, Mal and Banerjee proposed a seianalytical approach for the inversion of the spatial doain Fourier transfor, using the residue theore along one direction in the plane of the plate and then nuerically evaluating the integral along the other. However, that approach leads to a standing wave in the first direction, which is counter-intuitive. Phase velocity (x /s 8 6 A ode A ode S ode S ode Slowness (s/ 9. 6.8.6 5. 3. 8 33 A ode S ode Frequency-plate thickness product (MHz Aerican Institute of Aeronautics and Astronautics 7 Angle (degrees (a (b Figure. (a Dispersion curve (plot of phase velocity versus frequency for the first four Lab odes in an aluinu alloy plate (isotropic and (b Slowness surfaces (plot of inverse of phase velocity versus direction in a - thick unidirectional graphite-epoxy coposite plate at 5 khz (with graphite fibers along o /8 o. In coparison, less odeling work has been done for GW testing using structurally integrated piezos for SHM. Due to this, often little or no theoretical basis is provided by researchers for their choice of the various testing paraeters involved such as transducer geoetry, diensions, location and aterials, excitation frequency, bandwidth aong others. Aong the works that have sought to bridge this gap, Moulin et al. used a coupled finite eleent-noral odes expansion ethod to odel GW excitation in coposite plates with piezos. This was also a -D analysis. Soe researchers (Lin and Yuan, Rose and Wang 3, Veidt et al. have looked at using Mindlin plate theory for odeling GW excitation by circular and/or rectangular piezos in isotropic plates. That approach yields approxiate solutions and cannot odel higher GW odes. Giurgiutiu 5 studied the -D haronic excitation of Lab-waves in isotropic plates by infinitely wide surface-bonded piezos. As he suggested, the key difference 3
between NDE transducers and surface-bonded piezos is that the forer operate by tapping or causing noral traction on the surface, while the latter operate by pinching or causing shear traction at the actuator edges on the structural surface. Thus, the proble of odeling the 3-D GW field excited by finite diensional piezos in coposite plates has not received uch attention. C. Objective of this work In previous work by the authors, analytical solutions were presented for the proble of GW excitation and sensing by finite-diensional piezo transducers in isotropic plates using 3-D elasticity. The actuators were odeled as causing surface shear traction along their free edges, while the sensors were odeled as sensing the average in-plane extensional strain over their surface area. Those were validated with nuerical and experiental results. In the present paper, ultilayered coposite plates are addressed. The transducer odels are identical to that in the authors earlier work. The forulation and notation for the underlying ultilayered coposite plate is largely adapted fro Lih and Mal 9 and are suarized here for copleteness. A sei-analytical approach is used for the spatial Fourier inversion, which is distinct fro that in Ref.. The new approach in the present work yields traveling waves along both directions in the plane of the plate, as one would expect. II. Theoretical Forulation In this section, a general expression for the GW field excited by an arbitrary shape (finite diensional piezoactuator surface-bonded on a ultilayered coposite plate is derived. Consider an infinite N-layered coposite plate of total thickness H, with such an actuator bonded on one free surface, as illustrated in Figure. The origin is located on the free surface with the actuator and the X 3 -axis is noral to the plate surface. The individual layers are assued to have unidirectional fibers in a atrix and are odeled as being transversely isotropic with unifor density. This is a reasonable assuption if the GW wavelength is large copared to the inter-fiber spacing and the fiber diaeter. The solution procedure consists of the following four coponents (illustrated in Figure 3: (a First, one sets up the 3-D governing equations of otion for the bulk coposite ediu. The -D Fourier transfor is applied (or equivalently, plane waves propagating at a given angle in the plane of the fibers are assued. This yields the free-wave solution in ters of the eigenvectors and possible wavenubers through the thickness of the fibers. (b Then, one iposes the free-surface conditions of the plate along with the continuity conditions across interfaces (using the global atrix forulation. This also gives the allowable in-plane wavenubers for the possible GW odes. (c Next, the forcing function due to the presence of the surface-bonded piezo-actuators is iposed (assuing they cause shear traction along their free edges. This gives the solution in ters of a -D Fourier integral in the wavenuber doain. (d Finally, the -D wavenuber-doain Fourier integral is inverted (sei-analytically to yield the GW field due to haronic excitation by the piezo-actuator. The response to an arbitrary excitation wavefor can then be obtained by integrating the individual haronic coponents of the tie-doain signal (i.e., inverting the frequency-doain Fourier integral. Infinite coposite plate X Piezo actuator (odeled as shear distribution X H φ x x Figure. X 3 Infinite ultilayered coposite plate with arbitrary shape surface-bonded piezo actuator and piezo sensor 3 Aerican Institute of Aeronautics and Astronautics
X x (a Plane wave solution for infinite bulk coposite (Ref. 6 x 3 x O (b Free GW solution for infinite ultilayered coposite plate (Ref. 9 ˆ ˆ X 3 X R ˆ ˆ C (c Forcing function due to surface-bonded piezo (shear traction along edge I (d Wavenuber Fourier integral inversion (residue calculus Figure 3. Illustration of solution procedure Aong these, parts (a and (b are adopted fro Refs. 9 and 6. Part (c is based on the authors earlier work for isotropic plates,3. The details of the solution procedure are explained in the following sub-sections. A. Bulk waves in fiber-reinforced coposites First, consider the general solution for bulk waves in a transversely isotropic ediu. The equations of otion for the bulk ediu in each layer are: T c u = ρu ( where u is the displaceent vector, c is the stiffness atrix, the over a variable indicates derivative with respect to tie, ρ is the aterial density, and the operator is defined as: x x3 x = ( x x3 x x3 x x If the fibers are oriented along the -direction in the local coordinate syste (x, x, x 3 of the aterial, the stressstrain relation and the stiffness atrix c for a transversely isotropic aterial are: σ u, c c c σ u, c c c3 σ u 33 3,3 c c3 c c ;, with c3 = c c = c = (3 σ u + u 3,3 3, c σ u + u 3,3 3, c55 σ u + u,, c55 Here σ ij, with i and j taking integer values fro to 3, are the stress coponents. Next, constants are introduced that correspond to the squares of bulk wave speeds along different directions: Aerican Institute of Aeronautics and Astronautics
a = c / ρ (dilatational wave noral to the fiber direction a = c/ ρ (dilatational wave along the fiber direction a3 = ( c + c55 / ρ (shear wave in the plane of isotropy a = ( c c3 / ρ = c/ ρ (shear wave along the fiber direction a = c / ρ (shear wave in the plane of isotropy 5 55 Viscoelastic daping can be odeled by the use of coplex stiffness constants. Suppose the wavenuber coponents are ξ, ξ and ζ along the -, - and 3- local directions, respectively. Furtherore, without loss of generality, consider haronic excitation at angular frequency ω. Then the wave field is of the for: i( x x x3 t u = C e ξ + ξ + ζ ω (5 where C is a vector of constants. Then, fro Eqs. (-(5, one obtains the Christoffel equation: cξ + c55( ξ + ζ ( c + c55 ξξ ( c + c55 ξζ u u ( c c55 ξξ c55ξ cξ cζ ( c3 c ξζ u ρω u + + + + = (6 ( c + c55 ξζ ( c3 + c ξζ c55ξ + cξ + cζ u 3 u 3 For fixed values of ξ, ξ, and ω, there are six possible roots ±ζ i, i = to 3, of this equation. The first two pairs of roots correspond to pairs of quasi-longitudinal waves and quasi-shear waves 9,6. The wavenubers in the thickness direction for these four roots are, respectively: ζ = ξ + b ; ζ = ξ + b β β γ β β γ b = ; b α α = α α + α (7 α α = aa 5 ; β=( aa + a5 a3 ξ ω ( a+ a5 ; γ=( aξ ω ( a5ξ ω The third pair of roots corresponds to pure shear waves and their through-thickness wavenubers are given by: ζ3 = ξ + ( ω a5ξ / a (8 The displaceent eigenvectors resulting fro Eq. (6 corresponding to these roots are: T e = [ iξq iξq iζq] T e = iξ q iξ q iζ q (9 where e [ ] T [ iζ iξ ] = 3 3 q = a b ; q = ω a ξ a b q a b q a a b 3 5 = 3 ; = ω ξ 5 and the other three eigenvectors e, e 5 and e 6 are obtained by replacing ζ i by ζ i. The general solution for the displaceent vector is then given by: iζx3 iζx3 iζ3x3 iζx3 iζx3 iζ3x3 i( ξx+ ξx ωt u= ( C + ee + C+ ee + C3+ e3e + C ee + C e5e + C3 e 6e e ( where C i±, i = to 3, are free constants. B. Assebling the lainate global atrix fro the individual layer atrices With the general solution for the bulk ediu in place, one can then seek the particular solution for the proble at hand. As entioned earlier, the equations in this particular sub-section are fro Lih and Mal 9, and details can be found there. They are only suarized here. Due to the different orientations of the fibers in the different layers, it is useful to work with a global coordinate syste (X, X, X 3 distinct fro the local coordinate syste, for which the x -axis is aligned with the fiber direction. However, the X 3 - and x3 -axes are coincident and the two coordinate systes differ only in the plane of the plate. One can relate the displaceent vector u in the global syste and u in the local syste using the transforation atrix L (with the superscript indicating the layer nuber between and N, and φ being the angle between the X - and x -axes: ( ( 5 Aerican Institute of Aeronautics and Astronautics
cosφ sinφ u = L u where L = sinφ cosφ ( The surface traction conditions for this proble are: σ i3( X, X, = fi ( X, X, (3 N σ i3( X, X, H = ; i =,,3 where the functions f and f depend on the shape of the actuator and f 3 =. In addition, traction and displaceent continuity ust be aintained across the interfaces between the different layers. The -D spatial Fourier transfor is used to ease solution of this proble. For a generic variable ψ, it is defined by: and the inverse transfor is given by: ( + ψ Ψ (, = ( X, X e i X X dx dx ( i( X+ X = Ψ π ψ ( X, X (, e d d (5 Let U, Σ and F denote the -D spatial Fourier transfor of the variables u, σ and f, respectively. Furtherore, as for the bulk ediu solution, without loss of generality, haronic excitation at angular frequency ω is considered i t (the e ω factor is suppressed for convenience and is brought back at the end. Since continuity of both traction and displaceent has to be ensured across all interfaces, it is convenient to work with a displaceent-stress vector S in the transfored doain defined as: T S ( X3 = { U ( X3 Σi3( X3} (6 Then, fro Eqs. (3, (, and (: L Q Q E ( X 3 + C+ S ( X3 = ( X3 Q C (7 L Q Q E- ( X 3 C where: Q = e e e 3 ; Q = e e5 e 6 ρa5ξζ ( q + q ρa5ξζ ( q + q ρa5ξξ Q = ρaξζq ρaξζq ρa ( ξ ζ3 µ µ ρaξζ 3 ρa5ξζ ( q + q ρa5ξζ ( q + q ρa5ξξ Q = ρaξζq ρaξζq ρa( ξ ζ3 µ µ ρaξζ3 (8 T C = C+ C µ = ρ ( a5 a3 ξq ( a a ξq aζq µ = ρ ( a5 a3 ξq ( a a ξq aζq ( 3 3 ( 3 3 ( 3 Diag iζ X X iζ X X i E+ X = ζ 3( X3 X3 e, e, e iζ( X3 X3 iζ( X3 X3 iζ3( X3 X3 E ( X3 = Diag e, e, e with X 3 being the X 3 -coordinate of the interface between layers and (. Here Q and Q are atrices whose coluns are the stress eigenvectors for the th layer corresponding to wavenubers along the 3-axis, ζ i and ζ i, respectively. These are obtained fro the displaceent eigenvectors using Eq. (. The interface continuity conditions can then be expressed as: Q C = Q C where Q Q ( ; Q Q ( + + + + + + + = X3 + = X3 (9 6 Aerican Institute of Aeronautics and Astronautics
These equations ensure continuity of all displaceent and traction coponents at the interface across two layers. The surface traction conditions can be expressed as: ˆ ˆ N N Q+ C+ = F ; Q C = ( ˆ where ˆ N N N N N N Q + = LQ LQE ; Q = L QE L Q Here the atrices ˆQ correspond to the lower-half of Q relating to stress. The syste of equations is then solved by assebling Eqs. (9 and ( together into a 6N 6N banded atrix (called the global atrix, say G: Qˆ + C F Q Q+ C Q Q+ = + ( Q Q+ N N Q Q+ N ˆ N Q C Alternatively, if the layup is syetric about the id-plane of the plate, then the syste can be solved for the syetric and anti-syetric odes separately, thereby saving soe coputational tie. The surface condition ust also be split into its syetric and anti-syetric coponents. Then, the relevant surface condition on the top layer is enforced along with the continuity conditions up to the interface between layers N/ and N/ along with conditions of syetry (u 3, σ 3 and σ 3 being zero at the id-plane or anti-syetry (u, u and σ 33 being zero at the id-plane. The proble is thus reduced to two systes, each of coplexity 3N 3N. With the proble constraints now enforced, if the forcing function is also known, this equation can be solved to find the constants, C. C. Forcing function due to piezo-actuator The piezo actuator is odeled as causing in-plane shear traction of unifor agnitude (say τ per unit length along its perieter, in the direction noral to the free edge on the plate surface X 3 = (see Figure. In this odel, the dynaics of the actuator are neglected and it is assued that the plate dynaics are uncoupled fro the actuator dynaics. This odel was proposed by Crawley and de Luis 7 to describe quasi-static induced strain actuation of beas by surface-bonded piezo-actuators. For that case, they proved that the odel is a good approxiation if the actuator thickness is sall copared to that of the substrate and the bond layer is thin and stiff. This was also proven to be a good assuption in the authors earlier work for isotropic plates. In this work, two specific shapes of the piezo-actuator are considered: rectangular and ring-shaped. These are the ost coonly used shapes in GW SHM. For the ring-shaped actuator located at the center of the coordinate syste, the actuation coponents f i, i = to 3 and their respective -D spatial Fourier transfors F i are given by: f = ( δ( R Ao δ( R Ai cos Θ F = i( AoJ( Ao AJ i ( Ai f = ( δ( R Ao δ( R Ai sin Θ F = i( AoJ( Ao AJ i ( Ai ( f = F = 3 3 where A i is the inner radius of the ring-shaped actuator and A o is the outer radius. J ( is the Bessel function of the first kind and order one. δ ( is the Dirac-delta function. R and Θ are the polar spatial coordinates, i.e., ( R = X + X and tan ( X X and tan ( Θ= and and Γ are polar wavenuber coordinates, i.e., = ( + Γ=. Siilarly, for the rectangular actuator of diensions A A (along the X - and X -axes respectively, which is located at the center of the coordinate syste: f = τ( δ( X A δ( X + A ( He( X + A He( X A F = τsin( Asin( A/ i f = τ( He( X + A He( X A ( δ( X A δ( X + A (3 F = τsin( Asin( A/ i f = ; F = 3 3 7 Aerican Institute of Aeronautics and Astronautics
A A X 3 X X H A i Z A o R H (a (b Figure. Configurations considered: (a rectangular and (b ring-shaped actuators where He( is the Heaviside function. The constants in Eq. ( can then be analytically solved using Craer s rule: N (, C = Γ +, etc., (, Γ where N (, det F Q+ Q Q+ Γ = + Q Q+ Q (, Γ = det( G Q Qˆ N N + N ( D. Spatial Fourier integral inversion With the constants known, the expression for displaceent in the wavenuber doain can be obtained fro Eqs. (7 and (. The Fourier inversion forula, Eq. (5, can then be used to recover the spatial doain solution. For the case of a rectangular actuator, this leads to an expression of the following for for displaceent along the - direction in the spatial doain: π τ N(, Γ i( ( XcosΓ+ Xsin Γ ωt u( X, X, = sin( A cos Γsin( Asin Γ e ddγ (5 iπ (, Γ To solve the inverse Fourier integral, the residue theore fro coplex analysis is used for the integral along the - direction. For convenience, the integral in Eq. (5 is rewritten as: π i Acos Γ iacos Γ iasin Γ iasin Γ ( ( u ( X, X, = e e. e e τ N(, Γ (, Γ i( ( XcosΓ+ Xsin Γ ωt. e ddγ iπ (6 8 Aerican Institute of Aeronautics and Astronautics
τ = Γ+ π N(, Γ i( (( X Acos Γ+ ( X Asin Γ ωt u( X, X,. e dd iπ (, Γ π τ N(, Γ i( (( X Acos Γ+ ( X+ Asin Γ ωt. e dd (, iπ Γ π τ N(, Γ i( (( X+ Acos Γ+ ( X Asin Γ ωt +. e dd iπ (, Γ π τ N(, Γ i( (( X+ Acos Γ+ ( X+ Asin Γ ωt. e dd iπ (, Γ + Γ+ + Γ First, consider the first of the four integrals in the second line of Eq. (6, say I, which corresponds to (a, a. It is further rewritten as follows: Θ+ π τ N(, Γ i( R cos( Γ Θ ωt I =. e ddγ π iπ (, Θ Γ (8 X A where Θ = tan ; R = ( X A + ( X A X A This ensures that the coefficient of in the coplex exponential reains positive over the doain of integration. The inner integral along the real axis is replaced by a contour integral in the coplex plane, the sei-circular portion of which has radius (see Figure 5. The integrand is singular at the roots ˆ of (, Γ =, which is the dispersion equation for the ultilayered coposite plate. These roots are the allowable in-plane radial wavenubers for the ultilayered coposite plate at angular frequency ω. (, Γ is syetric about the -axis. Therefore, if ˆ is a root of (, Γ =, then so is ˆ. However, since only the outgoing wave is desired, the negative roots are not included in the contour. Using the residue theore for the inner integral in Eq. (6 yields in this case (assuing I is the integrand in I : ˆ ( ˆ Id + Id = π i Res I( C Γ + (7 (9 ˆ ˆ O ˆ ˆ R C I Figure 5. Contour integral in the coplex -plane to invert the displaceent integrals using residue theory 9 Aerican Institute of Aeronautics and Astronautics
It reains to be shown that the contribution fro C vanishes. As explained in Miklowitz 8 for a siilar plane wave excitation proble, ( N. (, Γ (, Γ for large is of order, and therefore tends to zero as. R I R I Furtherore, along C, if = i ;, > : Since I R and (. Therefore: R I I i( R cos( Γ Θ ωt iωt i R cos( Γ Θ R cos( Γ Θ R cos( Γ Θ e = e. e. e e cos( cos Γ Θ I are both always positive, the ter R Γ Θ C Θ+ π τ N( ˆ, Γ i( R ˆ cos( Γ Θ ωt I. e d ˆ ( ˆ π π Θ, Γ Aerican Institute of Aeronautics and Astronautics e ˆ ( ˆ Id = ; I = Id = π i Res I( = Γ (3 is finite and is bounded by zero as where ( indicates derivative with respect to. Siilar analysis can be used to solve the other three integrals in Eq. (6, to finally yield the expression for u : Θ+ π Θ + π τ N( ˆ, Γ ˆ i( R ˆ ˆ cos( Γ Θ ωt τ N(, Γ i( R cos( Γ Θ ωt u =. e dγ+. e d ˆ π ( ˆ Γ+, Γ ˆ π ( ˆ, Γ Θ π Θ π Θ 3 + π Θ τ ˆ N(, Γ + π i( R ˆ 3cos( Γ Θ 3 ωt τ. e d ˆ ( ˆ π π, Θ 3 Γ ˆ Θ π + Γ+ N( ˆ, Γ. e π ( ˆ, Γ i( R ˆ cos( Γ Θ ωt This procedure can also be used for the other displaceent coponents. An approxiate closed for solution can be obtained for the far field using the ethod of stationary phase. This is assuing daping is not odeled and that the integrand is real-valued. If daping is odeled, then a siilar approxiation can be done using the ethod of steepest descent 9. As explained in Graff 9, for large r: ψ ( i( r π irh ψ π ψ + f( ψ e dψ f( ψ e (33 rh ( ψ ψ Therefore, for large values of R (which leads to large values of R k, k = to : π τ ˆ N ( ˆ, Γ i( R cos( Γ Θ ωt+ π u = e + ˆ ˆ ˆ d ( cos( Γ Θ (, π R Γ. dγ Γ=Γ π τ ˆ ( ˆ N (, Γ i R cos( Γ Θ ωt+ π + e + ˆ ( ˆ cos( ( ˆ d Γ Θ, π R Γ. dγ Γ=Γ (3 π τ N( ˆ, Γ ˆ 3 i( R 3cos( Γ3 Θ 3 ωt+ π + e + ˆ d ( ˆ cos( Γ Θ 3 ˆ R π (, Γ3 3. dγ + ˆ d R. Γ=Γ3 π τ N ( ˆ, Γ e ( ˆ cos( Γ Θ ( ˆ, π Γ dγ Γ=Γ ˆ i( R cos( Γ Θ ωt+ π dˆ where tan( Γk Θ k = ˆ dγ. Thus, ( Γ k Θ k is the angle between the phase velocity and group velocity vectors 6. This iplies that the contributions fro Γ k doinate the integrals over Γ at large distances fro the source. This dγ (3 (3
reiterates a well-known fact about wave propagation in coposites, i.e., the wave travels at a steering angle, which ay be different fro the angle that it was launched along by its source. An analogous solution can be obtained for the ring-shaped actuators. In this case, an expression of the following for for displaceent along the -direction in the spatial doain is obtained: π τ N(, Γ i( ( XcosΓ+ X sin Γ ωt u( X, X, = ( AoJ( Ao AJ i ( Ai. e ddγ (35 (, π Γ For siplicity of analysis, consider the case of a circular actuator (A i =. The integral in Eq. (35 is rewritten thus: Θ+ π τ N(, Γ i( Rcos Θ ωt u( R, Θ, = AoJ( Ao. e ddγ π π (, Γ Θ (36 Θ+ π ( ( τ N(, Γ i( Rcos Θ ωt = Ao( H ( Ao + H ( Ao. e ddγ 8π (, Γ Θ π This is then re-arranged as follows: u ( R, Θ, = A H ( A N(, Γ i( Rcos Θ ωt. e ddγ+ (, Γ Θ+ π ( τ o o π 8π Θ Θ+ π ( τ o o π 8π Θ N(, Γ i( Rcos Θ ωt + AH ( A. e ddγ (, Γ As before, for each of the two integrals, residue calculus is used. The contours ust be chosen such that the integrands reain finite-valued. Over the sei-circular contour C: H A e H A e (38 ( i( Ao 3 π ( i( A 3 o π ( o ; ( o πao πao Therefore, for the first integral, over C, the integrand in the first integral of Eq. (37 is approxiately: i( ( Rcos Θ+ A 3 o π N(, Γ e π A (39 o (, Γ Therefore, the coefficient of in the exponent always reains positive and the contour can be closed in the lower half-plane over the full angular range, as was done for the rectangular actuator. However, for the second integral of Eq. (37, over C, the integrand is approxiately: i( ( Rcos Θ A 3 o π N(, Γ e π A ( o (, Γ In this case, the coefficient of in the exponent is negative over part of angular range (defined by cos Θ < Ao R. Therefore, over this part of the angular range, the contour ust be closed in the upper half of the coplex -plane. This results in a positive sign for the residue over that part of the angular range unlike earlier when closing in the lower half plane resulted in a negative sign for the residue. III. Ipleentation of the Forulation and Results for Haronic Excitation The theoretical forulation described above was ipleented in Fortran 9. The linear algebra package LAPAC for Fortran 9 was eployed to evaluate the deterinants of large banded atrices. The roots of the dispersion equation (, Γ, = were siply coputed by the zero-crossing approach, i.e., by evaluating the deterinant of the atrix over a fine grid in the (,Γ plane and looking for sign changes in the value of the deterinant. In doing so, one has to avoid the bulk wave velocities of the coposite aterial, which are also roots of the dispersion equation. One also has to take care to use double-precision variables and copute the roots with high precision since with the very large atrices involved, sall errors in the values of the roots cause large errors in the response solution. The code ipleented in Fortran 9 was coputationally efficient, with each run being copleted in a few inutes on a standard desktop coputer (. GHz Pentiu IV with 56 MB RAM. Soe results fro the proposed forulation are presented here. First, a unidirectional graphite-epoxy coposite plate (aterial properties: ρ = 578 kg/ 3 ; c = 6.73 GPa, c = 6. GPa, c = 3.9 GPa, c 3 = 6.9 GPa, c 55 = 7.7 GPa, this is the sae aterial used in Ref. 9 was analyzed. Daping was not considered for these analyses. The slowness surfaces (plots of inverse of phase velocity versus propagation direction for the A and S odes at Aerican Institute of Aeronautics and Astronautics (37
5 khz are shown in Figure (b. The out-of-plane haronic surface displaceent plots over a quarter section of this unidirectional plate at 5 khz due to a -c diaeter circular actuator surface-bonded at the center (S ode is shown in Figure 6. There is unifor actuation along all directions and the actuator diaeter is equal to the halfwavelength along the o /8 o direction, which is close to the optial size along that direction (actuator size optiization is discussed in ore detail for isotropic structures in the authors earlier work for isotropic plates. Despite this, and the fact that the predoinant direction of the group velocity vector is along the o /8 o direction, there is a strong preference for radiation along the 9 o /7 o direction, noral to the fiber direction. (The group velocity vector is noral to the slowness surface. This is logical, since the coposite plate is uch less stiff along this direction, thereby offering less ipedance to the GW field at 9 o /7 o copared to o /8 o. This is evident in the sei-log plot of the kernel function N/ versus the propagation angle (Figure 7, which relates to the excitability of the GW field along different propagation directions in the coposite plate. There is a difference of three orders of agnitude between the excitability along 9 o and o (the local valleys between 9 o and o are possibly due to structural anti-resonances. The directionality can only be weakly controlled by actuator design, as illustrated in Figure 8. In this configuration, the actuator diension along the o /8 o direction is still equal to the halfwavelength along this direction. However, the actuator diension along the 9 o /7 o direction is equal to two ties the wavelength along this direction, which nullifies the radiation eerging fro the actuator along it. However, the GWs launched at other angles tend to steer towards a direction close to 9 o /7 o. Siilar haronic plots for the A ode at 5 khz are shown in Figure 9 for a circular actuator and in Figure for a rectangular actuator. The diension of the circular actuator is optial approxiately at 8 o relative to the fiber direction, which results in the peak aplitude occurring roughly at that polar angle along the actuator perieter. However, again due to steering, the GW field is predoinantly along 9 o /7 o. The diensions of the rectangular actuator in Figure were again chosen to axiize radiation along o /8 o and iniize radiation along 9 o /7 o. Analysis was also done for plates with different quasi-isotropic layups of the sae graphite-epoxy aterial. The first layup was [/5/-5/9] s, with each ply being.- thick. For this configuration, up to around 5 khz, the S ode is practically isotropic. The slight variations in wavespeed with direction are iperceptible to the naked eye (e.g., see Figure. In addition, when excited by a circular actuator, the haronic GW field sees to be unifor across different directions (shown in Figure 3 for khz for the S ode in this quasi-isotropic layup. For the A ode at khz, the haronic GW field sees to be focused along two directions in the far-field (see Figure, which are the predoinant directions of the group velocity vector over the 36 o angular range. The layup [/9/5/-5] s was also analyzed for the A ode. The slowness surfaces of both layups are shown in Figure for the A ode (at khz. In this case, the slowness curve is closer to being isotropic and, therefore, the GW field tends to be ore unifor over the angular span in the far-field (see Figure 5. There is soe ild tendency of the group velocity vectors to be along the o /8 o and 9 o /7 o directions, which possibly explains the slightly stronger GW aplitude along these directions in the far-field. - - abs(n/ - -3 - Fiber direction 3 c Figure 6. Haronic radiation plot (u 3 over a quarter-section of a - unidirectional graphiteepoxy plate (S ode at 5 khz due to excitation by a -c diaeter actuator (lower-left corner, gray -5 3 5 6 7 8 9 Angle (degrees Figure 7. Variation of the kernel function N/ with propagation angle at 5 khz, S ode in the - thick unidirectional graphite epoxy plate Aerican Institute of Aeronautics and Astronautics
Fiber direction 3 c Figure 8. Haronic radiation plot (u 3 over a quarter-section of a - unidirectional graphiteepoxy plate (S ode at 5 khz due to excitation by a rectangular actuator of diensions a =.5 c, a = c (in gray, lower-left corner. Fiber direction c Figure 9. Haronic radiation plot (u 3 over a quarter-section of a - unidirectional graphiteepoxy plate (A ode at 5 khz due to excitation by a.3-c diaeter actuator (at the lower-left corner, in gray 9. 6.5 Slowness (s/ 5. 3 5e-5 8 33 Fiber direction c Figure. Haronic radiation plot (u 3 over a quarter-section of a - unidirectional graphiteepoxy plate (A ode at 5 khz due to excitation by a rectangular actuator of diensions a =.55 c, a =.36 c (in gray, lower-left corner. 7 Angle (degrees 3 Figure. Slowness surface for S ode in a quasiisotropic ([/5/-5/9]s, with each ply being.- thick graphite-epoxy coposite plate at khz. IV. Concluding Rearks This work addressed the odeling of the guided-wave (GW field excited by finite-diensional surface-bonded piezoelectric wafer transducers in ulti-layered coposite plates. The objective was to use the odels to support the design of GW structural health onitoring (SHM systes using this class of transducers for coposite structures. The individual layers were assued to be transversely isotropic, and the actuators were treated as causing shear traction along their free edges on the plate surface (assuing uncoupled actuator-substrate dynaics. The threediensional governing equations for the coposite substrate were used, and the global atrix approach was adopted to enforce surface and interfacial conditions, thereby capturing all possible GW odes without using reduced structural forulations. Expressions for the specific shapes of rectangular and ring-shaped piezo-actuators were derived. A rigorous Fourier inversion procedure was outlined to extract the outgoing wave solution. The nuerical ipleentation of the developed forulation was described. Saple results fro analysis done on soe configurations were then presented. For unidirectional coposites, it was observed that the direction noral to the fiber direction (9 o was the preferential direction of GW radiation, due to the lowest ipedance along it. By appropriately designing rectangular actuators, this tendency can be soewhat controlled close to the source. 3 Aerican Institute of Aeronautics and Astronautics
However, in the far-field, the GW tends to steer back to a direction close to the least ipedance one. For quasiisotropic layups, the S ode is approxiately isotropic at frequency-plate thickness products up to around 5 khz- while the A ode tends to steer towards the directions along which the group velocity vector is doinant. Future experients are planned with graphite-epoxy plates to further investigate the phenoenon. Slowness (s/ 5 9.5 6..5 3 8 [/5/-5/9]s [/9/5/-5]s 33 7 Angle (degrees 3 Figure. Slowness surfaces for A ode in graphite-epoxy plates with quasi-isotropic layups (with each ply being.- thick, at khz. o direction 5 c Figure 3. Haronic radiation plot (u 3 over a halfsection of the quasi-isotropic ([/5/-5/9] s graphiteepoxy plate at khz (S ode due to excitation by a.6-c dia. circular actuator (in gray. o direction c o direction c Figure. Haronic radiation plot (u 3 over a halfsection of the quasi-isotropic ([/5/-5/9] s graphiteepoxy plate at khz (A ode due to excitation by a.5-c dia. circular actuator (in gray. Aerican Institute of Aeronautics and Astronautics Figure 5. Haronic radiation plot (u 3 over a halfsection of the quasi-isotropic ([/9/5/-5] s graphiteepoxy plate at khz (A ode due to excitation by a.5-c dia. circular actuator (in gray. Acknowledgent This work is supported by the Space Vehicle Technology Institute under grant NCC3-989 jointly funded by NASA and DoD within the NASA Constellation University Institutes Project, with Ms. Claudia Meyer as the project anager. References Raghavan A. and Cesnik C.E.S., Review of guided-wave structural health onitoring, The Shock and Vibration Digest, Vol. 39, 7, pp. 9- Raghavan A. and Cesnik C.E.S., Finite diensional piezoelectric transducer odeling for guided wave based structural health onitoring, Sart Materials and Structures, Vol., 5, pp. 8-6 3 Raghavan A. and Cesnik C.E.S., "3-D elasticity-based odeling of anisotropic piezocoposite transducers for guided wave structural health onitoring," to appear in ASME Journal of Vibration and Acoustics (special issue on daage
detection and structural health onitoring, 7 Rose J.L., Ultrasonic waves in solid edia, Cabridge University Press, Cabridge, U, 999 5 Lab H., On waves in an elastic plate, Proc. Royal Society of London Series A, Vol. 93, No. 65, 97, pp. 93-3 6 Gazis D.C., Exact analysis of the plane-strain vibrations of thick-walled hollow cylinders, Journal of the Acoustical Society of Aerica, Vol. 3, 958, pp. 786-79 7 Ditri J., Rose J.L., Excitation of guided waves in generally anisotropic layers using finite sources, Journal of Applied Mechanics (Transactions of the ASME, Vol. 6, No., 99, pp. 33-338 8 Mal A.., Wave propagation in layered coposite lainates under periodic surface loads, Wave Motion, Vol., 988, pp. 57-66 9 Lih S.-S. and Mal A.., On the accuracy of approxiate plate theories for wave field calculations in coposite lainates, Wave Motion, Vol., 995, pp. 7-3 Mal A.. and Banerjee S., Guided acoustic eission waves in a thick coposite plate, Proceedings of the SPIE, Vol. 539,, pp. -5 Moulin E., Assaad J., and Delebarre C., Modeling of Lab waves generated by integrated transducers in coposite plates using a coupled finite eleent noral odes expansion ethod, Journal of the Acoustical Society of Aerica, Vol. 7, No., January, pp. 87-9 Lin X. and Yuan F. G., Diagnostic Lab waves in an integrated piezoelectric sensor/actuator plate: analytical and experiental studies, Sart Materials and Structures, Vol.,, pp. 97 93 3 Rose L.R.F. and Wang C.H., Mindlin plate theory for daage detection: Source solutions, Journal of the Acoustical Society of Aerica, Vol. 6, No.,, pp. 5-7 Veidt M., Liu T. and itipornchai S., Flexural waves transitted by rectangular piezoceraic transducers, Sart Materials and Structures, Vol.,, pp. 68-688 5 Giurgiutiu V., Lab wave generation with piezoelectric wafer active sensors for structural health onitoring, Proceedings of the SPIE, Vol. 556, 3, pp. - 6 Auld B.A., Acoustic fields and waves in solids Volues I & II, nd ed., R.E. reiger Publishing Co., Florida, 99 7 Crawley E.F. and de Luis J., Use of piezoelectric actuators as eleents of intelligent structures, AIAA Journal, Vol. 5. No., 987, pp. 373-385 8 Miklowitz J., The theory of elastic waves and waveguides, North Holland, New York, USA, 978 9 Graff.F., Wave otion in elastic solids, Dover Publications, New York,99 LAPAC user s guide 3 rd edition, Society for Industrial and Applied Matheatics, Philadelphia, 999 5 Aerican Institute of Aeronautics and Astronautics