Integral equation based modeling of theinteraction between piezoelectric patchactuators and an elastic substrateEvgeny Glushkov1, Natalia Glushkova1, Oleg Kvasha1 andWolfgang Seemann2
1 Kuban State University, PO Box 1581, Krasnodar 350080, Russia2 Karlsruhe University, PO Box 6380, Kaiserstraße 12, D-76128 Karlsruhe, Germany
Received 27 October 2006, in final form 10 February 2007Published 30 March 2007Online at stacks.iop.org/SMS/16/650
AbstractAn integral equation based model for a system of piezoelectric flexible patchactuators bonded to an elastic substrate (layer or half-space) is developed.The rigorous solution to the patch–substrate dynamic contact problemextends the range of the model’s utility far beyond the bounds ofconventional models that rely on simplified plate, beam or shell equations forthe waveguide part. The proposed approach provides the possibility to revealthe effects of resonance energy radiation associated with higher modes thatwould be inaccessible using models accounting for the fundamental modesonly. Algorithms that correctly account for the mutual wave interactionamong the actuators via the host medium, for selective mode excitation in alayer as well as for body waves directed to required zones in a half-space,have also been derived and implemented in computer code.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Recently, piezoelectric smart materials that produce astrong mechanical response to an applied electric fieldhave been gaining popularity. Electromechanical systemswith piezoelectric actuators and sensors in the form offlexible patches bonded to elastic waveguide structuresfind application in ultrasonic non-destructive evaluationand structural health monitoring, e.g. shell structures inaerospace units, active systems of vibration damping, precisionmechanical positioning gears and ultrasonic surface wavemotors etc. The piezoelectric patches are fabricated fromvarious materials, such as PZT ceramics, PVDF films, piezo-composite wafers and so on. Their advantages include lightweight, flexibility and relatively low cost. Therefore, inmany cases the patch actuators and sensors replace traditionalelectromechanical and piezo-crystal devices.
An important part of the design of piezoelectrically basedsystems is the development of mathematical models thatadequately describe elastic wave excitation by piezoelectricpatches. Traveling elastic waves are generated in the structure
Figure 1. Elastic waveguide with piezoceramic patch actuators.
by the contact shear tractions q resulting from longitudinalpatch deformation under transverse electric field Ez due tothe piezoelectric effect (figure 1). Being bonded to thestructure, the patches produce bending strains. Therefore,the first mathematical studies were devoted to flexural andlongitudinal wave motion of thin-walled structures governedby simplified beam, plate or shell equations [1–4] (see also
Integral equation based modeling of patch–structure interaction
surveys in [3, 5, 7]). In these models the actuating force isreplaced by a pair of opposing forces applied at the locationsof the patch edges (pin-force models). To produce action onthe waveguide, equivalent to the action of a modeled patch, theamplitudes of the forces are rigorously derived in terms of bothpatch and structure material and geometrical characteristics.
The beam and plate models operate with the firstfundamental (bending and longitudinal) guided wave modes,and so they provide a simple, physically clear description of theelastodynamic wave processes in the substructure. However,their application is restricted. They are valid only in a lowfrequency band where the characteristic wavelength is muchgreater than the plate or shell thickness. Therefore, modelsemploying the Rayleigh–Lamb equations for an elastic layerand half-space have been developing as well, in order totake into account higher Lamb modes or Rayleigh surfacewaves [5–9]. In most treatments, however, the action ofthe actuators on an elastic base is also replaced by a systemof forces that do not account for the mutual patch–structureinteraction, similar to the simplified models of the first group(plate and shell equations). Though such a simplification doesnot alter the eigen-characteristics of the excited modes, it limitsthe proper simulation of fine wave phenomena. First of all,if the loading is introduced without regard to the dynamicresponse of the structure, it is difficult to adjust the frequencydependence of energy characteristics, such as the total amountof wave energy radiated from the patches into the waveguide,as well as its distribution among the excited modes. In addition,the structure feedback is essential when a group of actuatorsis considered, e.g. for the development of mode selectionalgorithms (see section 5). Furthermore, a strict solution tothe contact problem yields the real, dynamic stress distributionin the patch–structure contact interface, which is of greatimportance for failure analysis.
The solution of a coupled problem is achievable usingthe FEM discretization technique. However, FEM solutionscannot provide a direct insight into the wave structure (e.g. togive at once the source energy distribution among severalguided waves excited in a multimode range). In addition, aclassical FEM is not applicable to infinite open waveguides,because it operates within spatially restricted discretizationdomains. This obstacle is not critical; it may be avoided byintroducing special infinite elements (e.g. the strip elementsproposed in [10]), or by using improved hybrid schemes, whichcombine FEM solutions in limited areas with expansions interms of guided waves going to infinity [11].
Meanwhile, it is possible to obtain a physically clearview of the wave structure with quantitative informationlike a FEM might provide, without invoking cumbersomehybrid procedures. This result can be accomplished viaan integral equation approach, based on the use of exactintegral representations of the wavefields generated in layeredelastic structures by surface tractions or buried forces. Thesubstitution of such integral expressions into the patch–layercontact boundary conditions reduces initial boundary-valueproblems to integral equations with respect to unknown contactstresses (in our case, to the shear traction q). Semi-analyticalsolutions of the integral equations enable fast wavefieldcalculations, especially by the use of the residual technique.The latter yields explicit expansions in terms of guided waves
obtained analytically as residues from the integrand’s poles.The information about the wave source (patch action) entersthe integral expressions with q, thus it is retained rigorously inthe residual expansions as well.
In this way the integral equation approach produces atool for both qualitative and quantitative fast parametricalanalysis of patch excited devices and systems. However,its practical implementation requires thorough preliminaryanalytical work. For this reason, its use in practical workis somewhat rare. Examples of such work include [12–15]dealing with a static statement [12], anti-plane shear wave [13]and Rayleigh wave [14, 15] excitation in an elastic half-space. The half-space Green’s matrix involved here in theintegral representations is much simpler in its form thanthe Green’s matrix for an elastic layer or layered structurerequired for modeling Lamb waves; consequently, we cannotcite any paper on the patch–layer interaction modeling by theintegral equation method, except the previous works of ourgroup [16, 17].
This research was started a number of years ago, followingfrom experience in deriving and implementing semi-analyticalsolutions to the integral equations that arise in elastodynamiccontact and diffraction problems. A variety of methods havesince been developed, such as expansion in terms of splines,orthogonal polynomials and radial basis functions, reductionto infinite algebraic systems, layered element discretizationand others (see, e.g., [18, 19] and surveys in [20, 21]).These methods seemed to be quite applicable for the patch–layer interaction analysis. In fact, no critical obstacles wereencountered while extending them to the problem currentlyunder consideration.
The integral equation approach proved to be a convenienttool for studying wave phenomena. It revealed, in particular,the nature of high-mode patch–layer resonance effects [16],and led to the development of algorithms for normal modeselective excitation and directional radiation [17]. Theprincipal objective of the present paper is to give a full andconsistent explanation of the model’s development. Numericalexamples that illustrate the model’s validity and capabilitiesare also presented. Wherever possible, we present numericalexamples different from [16, 17]. In particular, a newcomparison with a plate model is given in the validationsection 3, which is supplemented by examples of boundarycondition control, and by testing against the half-space modelfrom [14]. Some comparisons intended to estimate the limitsof capability of the second group of simplified models, whichrely on the layer equations but without strict patch–substratecoupling, are also discussed. New algorithms for body wavedirectional radiation control are presented in section 5, inaddition to the selective mode excitation algorithms developedin [17].
2. Mathematical model
The derivation of Green’s matrices and integral representationsis based on the application of the Fourier transform withrespect to horizontal space coordinates. Therefore, themathematical technique described below is applicable toany layered or otherwise vertically inhomogeneous (gradient)substructures having only plane horizontal boundaries and
interfaces (piece-wise continuous vertical stratification). Forexample, a pack of layers simulating laminate compositematerials is also possible. For clarity’s sake, however, wepresent the main idea using the example of a homogeneouselastic layer of finite thickness h. The peculiarities of themodel’s application to other kinds of elastic base are discussedin parallel, beginning with an elastic half-space having h = ∞.
2.1. Coupled patch–layer boundary-value problem
As a substructure, let us consider a homogeneous isotropiclayer of thickness h, occupying the domain −∞ � x �∞, −h � z � 0, with M thin and flexible stripactuators of thickness hm,m = 1, 2, . . . ,M , bonded to itstop surface z = 0 (figure 1). Specifically, but withoutloss of generality, suppose the actuators are fabricated frompiezoceramic material of class 6 mm poled in the z direction.Modeling piezoelectric films of other kinds should not requiremajor modifications.
The bottom surface z = −h is stress free, whilethe longitudinal deformation of the strips in response to atransverse electromagnetic field Eze−iωt causes unknown sheartractions τxz = qe−iωt on the otherwise stress-free top sidez = 0 in the disconnected region � = ∪m�m. �m :|x − xm| � am is the contact area of the mth strip. Thewaveguide is assumed to be under plane-strain conditions,so the load q generates a 2D in-plane harmonic wavefieldu(x, z)e−iωt , where u = {ux , uz} is the displacement vector(its complex amplitude). Below, the time-harmonic factor e−iωt
is conventionally omitted, and only space dependent complexamplitudes are shown.
Mathematically, u(x, z) obeys the Lame equations
(λ+ μ)∇div u + μ�u + ρω2u = 0 (2.1)
and the boundary conditions
τ |z=0 =[
q0
], τ |z=−h =
[00
]. (2.2)
Here λ,μ are Lame constants, ρ is the density and τ ={τxz, σz} is the stress vector at a horizontal surface element.Hereinafter, the vectors are assumed to be column vectors.To underline this fact their elements are written in braces, asdistinct from conventional brackets for matrices.
With ideally elastic material properties, boundaryconditions are to be supplemented by certain radiationconditions in order to assure uniqueness. As the radiationcondition we use the principle of limiting absorption, whichmeans that the solution for an ideally elastic medium is thelimit of the unique solution of the corresponding problem for amedium with attenuation ε as ε → 0.
The distribution of contact stresses q(x) should take intoaccount both the dynamic layer response and the longitudinalstrip deformation εx = ∂v/∂x (here v(x) is an unknownhorizontal patch displacement). Therefore, equations forthe longitudinal wave motion of the patches should also beintroduced into the mathematical statement. Henceforth, it isconvenient to consider q and v as a superposition of the contactstresses qm(x) and the displacements vm(x) of each strip:
q =M∑
m=1
qm, v =M∑
m=1
vm (2.3)
where qm(x) and vm(x) are extended on the entire axis x byzero:
qm(x) ={
q(x), x ∈ �m
0, x /∈ �m,
vm(x) ={v(x), x ∈ �m
0, x /∈ �m.
Due to the flexibility of the strips, we assume that theydo not produce vertical contact stresses σz , and vertical patchdisplacements do not influence q. We also assume thatthe patch thickness hm � λ, where λ is the characteristicwavelength for the patch, and as a consequence, v isindependent of the transverse coordinate z. In addition, theshear strains γxz are also supposed to be negligible. Theelectric field E = {0, Ez} is purely transverse and uniformwithin each strip (Ez = ∪m Ez,m , where Ez,m = |Ez,m |eiθm arecomplex constants having amplitudes |Ez,m | and phase shiftsθm for a driving field Ez interacting with each strip). Underthese assumptions, the generalized Hooke’s law, which givesa linear matrix relation between the components of the stressand strain tensors and the electrical field vectors E, is reducedto the scalar relationship [12]
σx = Ym
1 − ν2m
[εx − d31,m (1 + νm)Ez,m ] (2.4)
where Ym, νm and d31,m are the Young’s modulus, Poisson’sratio and electromechanical coupling constant of the mthpiezoceramic patch, respectively.
To derive the equation of motion for patches, let usconsider a strip element of length dx , height hm and widthdy (figure 1). The resultant longitudinal force d fx actingon this element is the sum of the forces associated with thestresses −σx and σx +dσx at the element’s side edges, the force−qm dx dy caused by the contact tension qm, and the inertiaforce vρmhm dx dy:
d fx = hm dy dσx − qm dy dx − vρmhm dy dx (2.5)
ρm is the material density; v ≡ ∂2/∂t2(ve−iωt ) = −ω2v.Taking into account (2.4), the equilibrium condition d fx = 0is reduced to the equations of patch motion
Mmvm ≡ 1
bm
(d2vm
dx2+ κ2
0,mvm
)= qm,
x ∈ �m, m = 1, 2, . . . ,M, (2.6)
where κ20,m = ω2ρmhmbm and bm = (1 − ν2
m)/(hmYm).The side faces of strips are stress free: σx (xm ± am) = 0.
In view of equation (2.4) this yields boundary conditions
dvm/dx |x=xm∓am = em,
em = d31,m(1 + νm)Ez,m, m = 1, 2, . . . ,M, (2.7)
for equations (2.6).The equations with respect to wavefield u and patch
displacement v are coupled not only by the same stressfunction q entering equations (2.2) and (2.6), but also by thecondition of displacement continuity
ux (x, 0) = v(x), x ∈ �. (2.8)
In total, the patch–layer electromechanical interactionis simulated by the solution to the coupled boundaryvalue problem with equations (2.1) and (2.6) and boundaryconditions (2.2), (2.7) and (2.8).
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Integral equation based modeling of patch–structure interaction
Figure 2. Integration contour �, poles ζk and cuts in the complexplane α.
2.2. Integral representations
The geometry of the problem allows us to apply the Fouriertransform with respect to the horizontal variable x :
F[u] ≡∫ ∞
−∞u(x, z)eiαx dx = U(α, z)
F−1[U] ≡ 1
2π
∫�
U(α, z)e−iαx dα = u(x, z).
(2.9)
Here F and F−1 denote the operators for the direct and inverseFourier transforms and � is an integration path going in thecomplex plane α along the real axis Imα = 0, deviating fromit to bypass real poles and branch points of the integrand U inaccordance with the principle of limiting absorption (figure 2).
In the Fourier-transform domain, the Lame equations (2.1)are reduced to a system of ordinary differential equations(ODEs), which may be written in the following matrixform [18]:
Y′(α, z) = A(α, z)Y(α, z). (2.10)
Here Y = {Ux ,U ′x ,Uz,U ′
z} is a column vector constructedfrom the components of the transformed displacement vectorU = F[u] and its z-derivative U′; A is a 4 × 4 matrix withelements easily derived from (2.1) in an explicit analyticalform.
The general solution to this ODE is
Y =4∑
j=1
c j m j eγ j z (2.11)
where γ j and m j are eigenvalues and associated eigenvectorsof matrix A:
γ j : det(A − γ j E) = 0 (2.12)
m j : (A − γ j E)m j = 0 (2.13)
while c j are unknown constants to be obtained from thetransformed boundary conditions (2.2). The roots of thecharacteristic equation (2.12) are derived analytically:
γ1,2 = ±σ1, γ3,4 = ±σ2;
σn =√α2 − κ2
n , n = 1, 2,
where κ1 = ω/vp and κ2 = ω/vs are wavenumbers of P and Sbody waves in the elastic volume; and vp = √
(λ+ 2μ)/ρ andvs = √
μ/ρ are their velocities. The branches of the squareroots σn in the complex plane α are fixed by the conditions
Re σn � 0 and Imσn � 0 as α ∈ �. The eigenvectors m j arealso obtained from (2.13) in closed form.
As a stress vector τ at an arbitrary surface element fixedby a unit normal n is connected with the displacement field uby the stress operator Tn, τ = Tnu ≡ λn div u + 2μ∂u/∂n +μ(n × curl u), the conditions (2.2) are reduced in the Fouriertransform domain to the matrix equalities
T Y|z=0 =[
Q0
], T Y|z=−h =
[00
](2.14)
in which
T =[
0 μ −iαμ 0−iαλ 0 0 λ+ 2μ
], Q = F[q].
Substitution of the general solution (2.11) into equations (2.14)yields a linear algebraic system
Bc = Q (2.15)
with respect to the vector of unknown constants c ={c1, c2, c3, c4}. The elements of matrix B are combined fromthe T and m j components, as well as the exponents e±σn h , inaccordance with equations (2.11) and (2.14).
In the case under consideration Q = {Q, 0, 0, 0}, whereasin the general case Q = F[q], where q = {q+,q−} includesboth loads q+ and q− applied to the surfaces z = 0 and z =−h, respectively. Consequently, with such general boundaryconditions the Fourier transformed solution U (Fourier symbolof u) may be written in the following form:
U(α, z) = K +(α, z)Q+(α)+ K −(α, z)Q−(α),−h � z � 0. (2.16)
The K ± are 2 × 2 matrices, the columns of which are theFourier symbols of displacements generated in the layer bytangential and vertical point-force loads, τxz = δ(x) for thefirst column, and σzz = δ(x) for the second one (δ is Dirac’sdelta-function). These loads are applied to the top or to thebottom surface for K + or K −, and denoted by Q± = F[q±].All of these solutions have the same general form (2.11), butthe constants c j are obtained from the four systems
B c = ei , i = 1, 2, 3, 4 (2.17)
which differ from (2.15) and from one another in the right-handsides only. Specifically, the ei are unit vectors having only thei th component non-zero. Thus, e1 = {1, 0, 0, 0} for the firstcolumn of K +, e2 = {0, 1, 0, 0} for its second column, and soon.
The explicit, analytical forms of the elements of Green’smatrices K ± may be found, for example, in [16, 21]. However,this is of little practical importance, because computercodes implementing their calculation may rely on algorithmsfollowing directly from equations (2.11), (2.17). In otherwords, with each new value of the input parameter α, thevalues of K ±(α, z) elements may be computed in accordancewith (2.11), where the c j are determined numerically from thesystems (2.17). Such a way of calculating Green’s matrices iseven better for coding than explicit expressions that are oftencumbersome.
Moreover, only this method enables practical computerimplementation for a multilayered structure. The substitution
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of general solutions written in the form (2.11) for each sublayerof an N -layered laminate into the external and interfaceboundary conditions leads to an algebraic system of size4N with respect to the vector of unknown constants c ={c1, c2, . . . , cN }. Here, the ck are vectors of length 4 for eachof the sublayers. Since only neighbor vectors ck and ck+1
are connected by matrix relations following from the interfacestitch conditions, the global matrix B of this system has ablock-diagonal structure. Hence, its solution may be organizedusing certain recurrent matrix algorithms going back to thepioneering algorithms by Haskel, Tompson, and Petrashen’(e.g., see the review of matrix algorithms in [22]). Anotherapproach is to obtain the solutions in each sublayer directly inthe form (2.16) with unknown interface stresses Q±, which areexcluded, then using the interface stitch conditions.
The algorithms developed for numerically stable Green’smatrix calculations are described in more detail in [18, 21].The main point is that with any stratification the solution isobtained in the form (2.16). Hence, as soon as the calculationof matrices K ± is implemented, the proposed general schemeof the coupled problem solution works, without regard to theirspecific appearance for the stratified substrate under study.
Returning to the problem at hand, since Q = {Q, 0, 0, 0}the wavefield u excited in the layer can be represented via theinverse Fourier transform in the following form:
u(x, z) = 1
2π
∫�
K1(α, z)Q(α)e−iαx dα (2.18)
where K1 is the first column of the matrix K + for the layer.For a laminate substructure, K1 is also the first column of K +,calculated in that case via the system (2.17) of a larger sizeproportionate to the number of sublayers, as explained above.
The simplest appearance of K1 is for a homogeneous half-space. In that case h = ∞ and the second of conditions (2.2)(at the bottom) is unnecessary. Instead, the radiation conditionsas z → −∞ lead to the requirement c2 = c4 = 0in (2.11). Consequently, only components c1 and c3 remainto be obtained from the system (2.17) reduced to the size 2×2.Obviously, with an N -layered pack lying on a half-space thissystem becomes of size (4N + 2)× (4N + 2).
2.3. Integral equations
The integral representation (2.18), being substituted into thecontact condition (2.8), reduces the initial boundary-valueproblem to the integro-differential problem
Kq = v
Mv = q,x ∈ � (2.19)
with 2M pointwise conditions (2.7) imposed on v(x). Here
Kq ≡∑
m
∫�m
k(x − ξ)qm(ξ ) dξ
= 1
2π
∑m
∫�
K (α)Qm(α)e−iαx dα (2.20)
is the Wiener–Hopf type integral operator. The Fourier symbolof its kernel K (α) = K (1)
1 (α, 0) is the first component of thevector-function K1 at z = 0. While the differential operatorMv generalizes the ODEs (2.6).
A natural way to solve this problem is Galerkin’sdiscretization scheme via vm(x) and qm(x) expansion in termsof certain basis functions (splines, orthogonal polynomialsand so on). Its main advantage is in comparatively simplecomputer implementation regardless of the specific propertiesof the substructure (it may be either a layered waveguideor a half-space). On the other hand, with an increase infrequency, the form of the approximated functions becomesmore complicated (e.g., see v(x) in figure 4 below), whichrequires supplementing the number of terms, and so enlargesthe size of the systems and the computational costs.
Another efficient approach relies on a path integralevaluation using the residual technique. With a puremeromorphic K (α) (that is the case with any layeredsubstructure of finite thickness h) the problem reduces toan infinite algebraic system with regard to the unknowncoefficients t±
m,k in the wavefield expansion in terms of normalmodes (see (2.22) below). This method is more difficultto implement than the Galerkin scheme, because it requirespreviously finding all real and complex poles ζk and zeroszl of the Fourier symbols of the kernels in the complexplane α. On the other hand, it provides higher accuracy andlower computational costs, by taking explicitly into accountthe guided wave structure of the solution, which is obtainedin a physically clear analytical form. In other words, thecorrugation of the sought functions q and v due to thespatial oscillation of interface waves in the contact zone isexcluded from the approximation and from the numericalcomputation. Only the coefficients that are not sensitive tofrequency expansion are found. As a result, this approachhas little sensitivity to increasing frequency. Therefore, thisis the basic method that we use for modeling the patch–layerelastodynamic interaction.
We should remark that terms related to the complexbranches of dispersion curves cannot be neglected here asin a conventional guided wave analysis, because the seriesrepresentations in terms of residuals (like (2.22) below) areemployed in the near field, where the exponential evanescencecaused by complex orders does not hold. Instead, specialmethods for the stable truncation of infinite systems, basedon information about the asymptotic distribution of poles andzeros in the complex plane, have been developed.
The general scheme for the case of one surface obstacleis given in [19], while the modification for the case underconsideration, with several strips bonded to a layer, ispresented in [16]. The most detailed description is alsoavailable in the report [23]. Therefore, in the present paperwe shall restrict ourselves to the remarks above, consideringq and v in the subsequent text as known functions obtained inone way or another from the coupled problem (2.19), (2.7).
2.4. Wavefields in the substructures
As soon as K1 and the unknown contact traction q are obtained,equation (2.18) provides a way to calculate the wavefield uexcited in the substructure. In view of (2.3), Q = ∑
m Qm withQm = F[qm], and the total wavefield can be represented as thesuperposition of the wavefields um generated independently byeach load qm: u = ∑
m um. The fields um are written in thesame form as (2.18) with Q replaced by Qm.
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Integral equation based modeling of patch–structure interaction
It is notable that the um are not the fields generatedby solitary strips independently of all other actuators, as isconventionally assumed in simplified models. The mutualwave interaction of the actuators via the substructure resultsin changes in the contact stresses qm(x), which may differconsiderably from the stress under a single strip. In the modelpresented, the mutual influence of actuators working in a groupis strictly taken into account in the integral relationship (2.19),which connects the qm with each other.
Furthermore, with any laminate waveguide of finitethickness h < ∞, including, of course, the homogeneouslayer under study, the elements of matrices K ±(α, z) aremeromorphic functions of α (i.e. without branch points). Inthe complex plane α they have a finite number of real poles±ζk, k = 1, 2, . . . , Nr and an infinite set of complex ones:±ζk, k = Nr + 1, Nr + 2, . . .. These poles are the roots of thecharacteristic equation
�(α,ω) ≡ det B(α, ω) = 0 (2.21)
where B is the matrix in systems (2.15) and (2.17). The polesappear in pairs, because �(α,ω) is even with respect to eitherof the variables. We assume that the poles +ζk are located inthe α-plane above the integration path �, while −ζk lie belowit, and the complex poles are arranged in order of imaginaryparts increasing (| Im ζk+1| � | Im ζk|) (figure 2).
The meromorphic property permits each of the fields um
excited outside the contact domain �m to be represented interms of normal modes:
um(x, z) =∞∑
k=1
t±m,k a±
k (z)e±iζk (x−xm∓am), x /∈ �m.
(2.22)Here t±
m,k = i rk Qm(∓ζk)e±iζk(xm±am), rk = res K (α)|α=ζk ,a±
k (z) = K1(α, z)/K (α)|α=∓ζk ; the upper and lower signs foreach item are taken for x > �m and x < �m, respectively.The terms corresponding to real poles ζk describe guided wavesgoing to infinity with phase and group velocities cp,k = ω/ζk
and cg,k = dω/dζk , while complex ζk yield inhomogeneous,exponentially decaying waves.
The functions Qm(α), representable via their Fourierintegrals over the limited intervals |x − xm| � am, are entirefunctions. In other words, they have no poles and no branchpoints in the whole complex plane α. That is why there are noresiduals of Qm in the series (2.22). Their exponential behaviorat infinity, Qm(α) ∼ O(ei(xm±am)α), as α → ∞, is governed bythe contribution of the edges x = xm±am to the asymptotics ofthe oscillating Fourier integrals F[qm]. Furthermore, in viewof the structure of the solution of the coupled equations (2.19)in the Fourier transform domain (e.g., see (3.7) in [16]), we canwrite them in the following general form:
Qm(α) = (C−m eiαam + C+
m e−iαam)eiαxm/(K − bmGm) (2.23)
where the C±m (α) are unknown functions having non-
exponential behavior at infinity. Gm = 1/(−α2 + κ20,m)
is the Fourier transform of the fundamental solution toequation (2.6).
Though the C±m (α) may have poles, the latter must be
eliminable in the functions Qm as whole. An additional set
of eliminable poles ±zl , l = 1, 2, . . ., is the roots of thecharacteristic equation
K (α) − bmGm(α) = 0 (2.24)
(we assume ±zl are arranged in the α-plane in the same wayas the poles ±ζk above). Their eliminability is assured by theconditions
C−m (±zl)e
±izl am + C+m (±zl)e
∓izl am = 0, l = 1, 2, . . .(2.25)
with which any C±m entering (2.23) must comply. In our
approach such conditions are used for generating infinitealgebraic systems, to which integral equations with ameromorphic K (α) are reduced.
Representation (2.22) is not valid for the wavefield inthe contact zone x ∈ �m, because it is derived under theassumption |x − xm| > am, which makes the exponentialfunction e−iαx in the integrand (2.18) principal. It determinesthe direction of the contour closing into the upper half-planeIm α � 0 if x < xm − am, and conversely into the lower half-plane if x > xm + am, in accordance with Jordan’s lemma.
As x ∈ �m (|x − xm| < am), the exponentials eiα(xm±am)
entering Qm(α) become the governing ones. Consequently, theintegrand is split into two parts in accordance with the directionof closing (upward for the terms with exponents eiα(xm+am) anddownward with eiα(xm−am)). By doing so, the eliminable poleszl of the entire functions Qm(α) become non-eliminable, whilethe poles ζk of K (α) in (2.18) are, in contrast, being removed.Therefore, the wavefield in the contact zone�m, as well as thecontact stress functions qm(x), are expressed in the followingseries form:
u(x, z) =∞∑
l=1
[s+m,lb
+l (z)e
izl (x−xm+am)
+ s−m,l b
−l (z)e
−izl (x−xm−am)], x ∈ �m
qm(x) =∞∑
l=1
[s+m,l e
izl(x−xm+am) + s−m,l e
−izl(x−xm−am)],
x ∈ �m
s±m,l = ∓iC±
m (∓zl)res (1/(K − bmGm))|α=∓zl ,
b±l = K1(∓zl , z).
(2.26)
For real zl the terms with exponential factors e±izl x describeguided waves propagating in the contact area from the edgesxm ∓ am in the right and left directions, respectively. Thus,the zl are the wavenumbers for interface waves excited in thepatch–layer composite structure.
As an example, several first real branches of the dispersioncurves ζk(ω) and zl(ω) are displayed in figure 3 (hm =h/6). One can see that the first two curves z1(ω) andz2(ω) are practically the same as the curves relating to thefundamental antisymmetric and symmetric modes a0 and s0
(poles ζ1 and ζ2, respectively). This means that due to patchflexibility and comparatively small thickness its presence isnot distinguishable in the characteristics of the fundamentalmodes, whereas the curves of higher modes differ visibly.
It is appropriate to mention here that throughout the paperwe present numerical results in a dimensionless form fixed by
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Figure 3. Normal mode wavenumbers ζk, zl and ξm versus ω(dispersion curves) for elastic layer, patch–layer composite andsimplified plate waveguides. The principal patch–layer resonancemodes are marked by circles.
several basic units. For a homogeneous elastic layer we takel0 = h as the unit of length, v0 = vs as the unit of velocity andρ0 = ρ as the unit of material density, where h, vs and ρ arelayer thickness, S-wave velocity and density of layer material.The dimensionless angular frequency ω = 2π f h/vs , where fis the dimensional frequency. With a half-space (h = ∞),the unit of length is usually taken so as to keep the samedimensionless size of patches as with a layer; e.g., l0 = 6hm tobe comparable with the examples in figure 3 and below.
Unless otherwise specified, the numerical examples in thispaper use the following dimensionless input parameters.
Substrate: vs = 1, ρ = 1, h = 1 (layer) or h = ∞(half-space), ν = 0.3, Y = 2.6
These relate to the case of ST-37 steel as the substrate andCeramTec Sonox P88 piezoceramic actuators with materialproperties s E
11 = 14, 3 × 10−12 m2 N−1, s E12 = −4, 9 ×
10−12 m2 N−1, ρ0 = 7, 83 × 103 kg m−3, and d31 = −135 ×10−12 C N−1. Since the patch material is anisotropic, weassume Ym = 1/s E
11 and νm = −s E12/s
E11.
The input data em = 1 mean that the results are normalizedin addition to the dimensional value em that enters the edgeconditions (2.7) (e.g., in figures 4, 6 and 7 |ux | := |ux |/he1).Even so, specific values of the electromechanical constants d31
and driving fields Ez are not fixed.
If a substructure has infinite thickness (a layered half-space), the elements of the Green’s matrices are not puremeromorphic, because of the branch points ±κn, n = 1, 2that appear in the α-plane in addition to the poles (here κn
are wavenumbers of body waves for the underlying half-space;see notations after equations (2.11)–(2.13) above). Therefore,after closing the contour, the path integrals over the cut sidesremain as an addition to the sum of the residues (2.22). The far-field asymptotics of these integrals yield P and S body wavesexcited in the underlying half-space.
In the simplest case of a homogeneous half-space, the far-field wave structure takes the form
Figure 4. Test of the contact boundary condition (2.8). Solid linesand dots are markers for u(x, 0) and v(x), respectively.
um =
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
t±m,Ra±
R e±iζR(x−xm∓am)(1 + O(|λ/|x |),|x/λ| → ∞, z = O(1)
2∑n=1
t(n)m (ϕ)eiκn Rm/√
Rm/λ (1 + O(λ/Rm),
Rm/λ → ∞, −π < ϕm < 0.
(2.27)
Here λ is the characteristic wavelength at a fixed frequencyω, ζR = ζ1 is the only real pole (the Rayleigh pole) and t±
m,R
and a±R have the same forms as t±
m,1 and a±1 in (2.22) above.
Hence, the first line of equation (2.27) yields the Rayleigh waveexcited by the mth strip, while the third line gives P and Sbody waves written in polar coordinates (Rm, ϕm) centered inthe middles of the strips: x − xm = Rm cosϕm, z = Rm sinϕm.The amplitude vectors of these waves
t(n)m = −√κn
2π isinϕmK1,n(αn)Qm(αn)e
−iαn xm
are derived by the steepest descent method as contributions ofthe stationary points αn = −κn cosϕm; K1,n : K1(α, z) =∑2
n=1 K1,n(α)eσn z .
2.5. Energy representations
The flow of wave energy in a time-harmonic field is specifiedby the energy density vector e = {ex , ez} time-averaged overthe period of oscillation 2π/ω (the Umov–Poynting vector).The total amount of energy E carried by harmonic wavesthrough a surface S (a time-averaged power) is obtained bysurface integration
E =∫
Sen dS (2.28)
where en = e · n = −ω2 Im(u, τ n), n is the outward unit
normal to S at a current point of integration. τ n = Tnu isthe stress vector at this point of S with a 2D in-plane motion(u, τ ) = uxτ
∗x + uzτ
∗z . Hereinafter, an asterisk denotes
complex conjugation.Specifically, the source energy E0 coming from the
actuators into the substrate can be obtained by integration overthe surface z = 0. Due to the boundary conditions, it takes the
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Integral equation based modeling of patch–structure interaction
Figure 5. Comparison of results from [14] for a half-space substrate. Quasistatic contact shear stress distribution q(x) under a single actuator(a), and stress distribution τx z at the depth z = −0.5a caused by two actuators (b).
form
E0 = ω
2Im
∫�
v(x)q∗(x) dx
= ω
4πIm
∫�
K (α)Q(α)Q∗(α∗) dα. (2.29)
For the energy fluxes E± outgoing to infinity through verticalcross-sections x = ±x0 equation (2.28) yields
E±(x0) = ∓ω2
∫ 0
−h(u(±x0, z), τ x (±x0, z)) dz (2.30)
where τ x = Tx u = {σxx , τxz} is the stress vector at an areaelement with the normal n = {1, 0}.
If the substructure is a layer (h < ∞), K (α) is real withreal α, so that only residues from real poles ζk contribute to theimaginary part of the path integral in (2.29):
E0 =Nr∑
k=1
(E+k + E−
k )
E±k = ω
4res K (α)|α=ζk |Q(∓ζk)|2
= ω
4rk
∣∣∣∣∣M∑
m=1
t±m,ke∓iζk (xm±am)
∣∣∣∣∣2
.
(2.31)
Substitution of u, taken in the form of normal modeexpansion (2.22), into equality (2.30) leads to the sameexpressions as appear in (2.31), which confirms the energyconservation law for a layer: E0 = E+ + E−. Besides, itmeans that E±
k are energies carried to infinity to the right andleft of the sources by each of the traveling waves associatedwith the real poles ζk .
With a homogeneous half-space (h = ∞) the presence ofbranch points κn results in Im K (α) �= 0 at |α| < κ2, so thatinstead of (2.31) the representation of source energy takes thefollowing form:
E0 = ER + Ev ER = E+R + E−
R ,
Ev = ω
4π
∫ κ2
−κ2
Im K (α)|Q(α)|2 dα(2.32)
here E±R = E±
1 are contributions of the Rayleigh poles±ζ1, which coincide with the energy outflow to infinity in ahorizontal direction calculated by the integration (2.30). Therest of the source energy Ev , obviously, goes out through the‘bottom’ z = const, being carried by body P and S waves.
This is confirmed by numerical integration over a horizontalplane |x | < x0, z = const as z → −∞ or over a cylindricalsurface x = R cosϕ, z = R sinϕ, −π � ϕ � 0, as R → ∞.
Explicit expressions for wave energy fluxes are usedfurther on as objective functions in the algorithms developedfor selective mode excitation and directed radiation.
3. Validation
There are two aspects in the validation of the developed model.First, we have to make sure that the code yields numericalsolutions appropriate to the boundary-value problem posed insection 2.1. Second, we should test it against other, existingmodels, and thereby estimate the range of their practicalcapability.
The first task is accomplished by verifying whether acalculated wavefield u obeys the governing equations, andsatisfies the boundary conditions. It should be noted thatdue to the semi-analytical structure of the solution, it obeysidentically equations (2.1) and conditions (2.2) with any q, assoon as Green’s matrix K (α, z) is derived correctly. Therefore,the main issue is control of the bonding condition (2.8) thatholds if the solution to the coupled problem (2.19) is correct.
Examples of such tests at various frequencies ω aregiven in figure 4 for two patches having the above selectedparameters, centered at x1,2 = ±8.333, and actuated witha phase shift θ = π/2 (e1 = 1, e2 = i ). One cansee that even at high frequencies, when the form of contactdisplacements becomes very complicated, ux (x, 0) calculatedvia (2.1) coincides with v(x) very well. Besides, the figuresdemonstrate how difficult it is to guess the proper form for thecontact tensions for uncoupled modeling of the actuator actionat higher frequencies.
In addition to verifying the boundary conditions, theenergy conservation equalities, as well as stitching ofasymptotics (2.22), (2.27) with the near field integralrepresentation (2.18), are also used as ongoing controls.
As for comparisons with other integral equation basedmodels, figure 5 presents plots taken from [14] (solid linesfor the results obtained by the method developed in that paperand round circles for FEM results) being superimposed onour results given by dots (both results are in the unifieddimensionless form). The left subplot is for the contact stressq(x) under the right half of a single strip of semi-width a = 1bonded to an elastic half-plane (h = ∞). The right subplotis for the distribution of shear stresses τxz produced by two
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Figure 6. Comparison of layer and plate models in the frequencydomain. Surface displacement |ux (x, 0)| versus ω for two pointsx = 1 (between the actuators (a)) and x = 50 (far away from them(b)).
actuators in the substrate at depth z = −0.5. One can see novisible difference among the solutions to equation (2.19) (dots)and to the integral equation [14] (solid lines), while the FEMresults deviate slightly from them.
This comparison is made for a quasistatic state (ω → 0).For moderate dimensionless frequencies ω > 0 a reasonableagreement of the results also holds, whereas as ω increases thediscrepancy becomes increasingly large, up to a considerabledifference at ω > 10, when the contact stress becomes heavilywaved. The reason for this discrepancy is, obviously, infreezing in [14] the number of Chebyshev polynomials (termsof discretization) used in the quasistatics for the study at allfrequencies, whereas, as underlined in section 2.3 above, toretain accuracy, the number of basis functions in Galerkin’sscheme has to be increased in parallel with the frequencyincrease. The residual technique approach to the solution ofthe contact problem developed in [19, 16] is based on explicitlyaccounting for interface waves in the contact zone, so it is muchless sensitive to frequency increase.
Comparisons have been carried out for a layer model (h =1), first against beam [24] and plate [16] models approximatinglayer behavior at low frequencies ω � 1. As expected,these results agreed well in the low-frequency band ω < 0.1,but were quite different for ω > 1, even in the subrange1 < ω < 3.14 where only two fundamental modes (bendingand longitudinal) exist in both the layer and plate waveguides(see figure 3).
These comparisons were made for vertical displacementsw of the middle plate or beam surface governed by the bendingwave equation
d4w
dx4− ξ 4
1w = 0. (3.1)
Further, figures 6 and 7 present similar test comparisons for thehorizontal displacements ux(x, 0) that depend on both bendingand longitudinal waves:
ux (x, z) = u(x)− (z + h/2)w′(x). (3.2)
In the context of Kirchhoff’s plate theory w(x) is as above,and u(x) is the axial displacement of the middle plate surface
Figure 7. Spatial distributions of displacements |ux (x, 0)| from layer(solid lines) and plate (dashed lines) models at low frequencies.
z = −h/2 governed by the equation
d2u
dx2+ ξ 2
2 u = 0. (3.3)
In these equations ξ 41 = ρω2h/D and ξ 2
2 = ρω2h/Bare the wavenumbers for bending and longitudinal waves inKirchhoff’s plates, respectively (in figure 3 they are depictedby chain lines). D = Y h3/12(1 − ν2) and B = Y h/(1 − ν2);Y and ν are, respectively, Young’s modulus and Poisson’s ratiofor the plate material.
In accordance with the pin-force model [2], the action ofthe actuators on a plate is simulated by pairs of antiphase forces∓Fm applied at the end points of strips x = xm ∓ am. Specificvalues of the point forces Fm may be derived in different ways(e.g., see the survey in [3]). Following [2], we arrived at Fm =12em D/h2(4+ψm), where ψm = Y h(1− ν2
m)/Ymhm(1− ν2).Figure 6 gives frequency dependences for the amplitudes ofhorizontal displacements ux(x, 0) at the surface of the layer(solid lines) and the plate (dashed lines) computed usingequations (2.18) and (2.19) and (3.2), respectively, for twoactuators centered at x1,2 = ∓2am,m = 1, 2.
Like the examples given in [16, 24], one can see very goodagreement in the band ω < 0.1, and different behavior aboveω = 3.14, where higher modes appear in the layer in additionto the pair of fundamental modes. Certain (non-crucial)differences in an intermediate range (approximately from ω =0.1 to 0.5) arise, because the quasistatic approximation usedfor the pin forces Fm does not take into account the inertialproperties of the patch–plate composite. In addition, at higherfrequencies the curves for the plate fundamental modes ξ1 andξ2 deviate from those for the layer or patch–layer waveguide.The deviation of ξ1 from the a0-mode curve ζ1 is apparent infigure 3 for ω > 0.5, while ξ2 differs considerably from thes0-mode curve after ω > 3 only where the curve ζ2 makes aturn.
The plots of figure 7 confirm that at low frequenciesω < 0.1 the plate model yields the correct spatial displacementdistribution as well.
From these examples one can conclude that theapplicability of beam and plate models is limited to a rathernarrow low-frequency band, which is much less than the two-mode range 0 � ω � π . In the example considered, they are
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Integral equation based modeling of patch–structure interaction
Figure 8. Total energy E0 radiated by two patches (a) and itspartition Ek/E0 among the Lamb modes (b)–(d). Solid lines are forthe present coupled model; dashed and dotted ones are for theconstant pin force and frequency dependent actuating in accordancewith [6] and [5], respectively.
valid for ω < 0.1 or λs > 20πh, where λs = vs/ f is theS-wave length in the plate.
One might expect a wider range of application for thesecond group of simplified models, which rely on expansionin terms of Lamb modes but without rigorous patch–layercoupling. Indeed, the plots in figure 8 confirm suchexpectations, revealing very good agreement at least in the firsthalf (ω < π/2) of the two-mode range. This figure shows thefrequency dependences for the total energy E0 radiated by twopatch actuators (subplot (a)), and its partition Ek/E0 among theexcited guided waves (subplots (b)–(d)). The values obtainedvia the coupled problem are plotted by solid lines, while thedashed lines are for the pin-force sources
q0 is a normalizing factor fixed in order to adjust the curves E0
at low frequencies. In addition, the dotted line in subplot (a)is for E0, radiated by the frequency dependent uncoupledexcitation forces, derived in [5] for patch actuators attached tobeams. In these formulas we have replaced the beam structuralstiffness by the layer structural stiffness without any forcenormalization. However, this model gives very large deviationin the second half of the two-mode range, even in comparisonwith simple, constant pin forces (3.4).
Since the uncoupled layer models also operate withLamb guided waves, the only difference from the presentcoupled model is in the form of the contact stresses q(x)entering the wavefield representations (2.18) and (2.22) via itsFourier transform Q(α) = ∑
m Qm(α). Consequently, thedeviation of the curves in figure 8, which becomes noticeablestarting from the second half of the two-mode range, resultssolely from the distinction between the Fourier transformsQm(α) = 2iq0eiαxm sinαam for the prescribed point-forcesources (3.4), versus the correct Qm provided by the coupled
Figure 9. Illustration of the mechanism of strip size resonance.
model. The deviation increases at higher frequencies, due tothe considerable complexity of the contact characteristics withfrequency increase (see figure 4).
These examples allow us to conclude that our novelapproach extends the capability of computer simulation farbeyond the limits of conventional models. As for the limitsof the coupled model itself, in principle, problem (2.19) canbe properly solved at rather high frequencies. However, thepractical limits are narrower, due to the assumptions leading tothe 1D equations (2.4)–(2.7). These limits might be estimatedby comparison with experimental measurements and/or FEMcomputations.
4. Resonance properties of wave energy radiation bya piezoceramic strip actuator
4.1. Plate and layer models
With fixed material and electric field parameters, the energyE0 (2.29) supplied by a single strip depends only on thefrequency ω and the strip width 2a. The dependencesE0(a) with ω = const feature periodicity due to alternatecomposition of waves coming from strip edges arriving inphase and out of phase. The mechanism for this effect is seenmost clearly in the low-frequency band, where the pin-forceapproximation for the strip action is valid.
Since the point forces Fm and moments Mm = Fmh/2that represent the strip action operate in antiphase, the excitedwaves combine in phase when the distance 2a between theedges is equal to a half-integer number of their wavelengthsλ (figure 9):
2a = λ(n + 1/2), n = 0, 1, 2, . . . .
For bending (m = 1) and longitudinal (m = 2) modes,the wavelengths λ = 2π/ξm (ξm are introduced after (3.3)).Therefore, the strip sizes an , providing maximal amplitude ofone of the excited plate modes, are determined by the simpleformula
an = π(n + 1/2)/ξm, n = 0, 1, 2, . . . ;m = 1, 2. (4.1)
However, in a layered waveguide these simple relationsbecome inapplicable as soon as ξm differs from thewavenumber zl of the related interface fundamental mode inthe patch–layer composite. Obviously, in this case ξm in (4.1)must be replaced by zl .
Indeed, the values of an shown in the subplots (b) and (c)of figure 10 by vertical lines coincide well with the maximaof each mode contribution Ek(a) in the source energy E0 =E1 + E2. E0(a) is given in the upper subplot (a). The
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E Glushkov et al
Figure 10. Patch size resonance: radiated energy E0 versus striphalf-width a. The vertical lines denote the valuesan = π(n + 1/2)/zl for zl taken from the principal branches markedin figure 3 by circles.
formulas (4.1) of plate theory would yield different values an ,being unmatched at this frequency ω = 1 with Ek maxima.They would be even more incorrect at higher frequencies.
Numerical analysis has shown that in the multimode rangeω > π , as a rule, only one of the wavenumbers zl determinesthe resonance sizes an = π(n + 1/2)/zl , n = 0, 1, 2 . . .,while the others do not contribute noticeably to the E0(a)dependences. In figure 3, such principal sections of dispersionbranches zl are marked by circles (l = 3 for 2.7 < ω < 8.5,l = 4 for 8.5 < ω < 14, l = 5 for ω > 15 and so on). It isinteresting that these sections are aligned in a single line. Thedispersion curve closest to this line is the curve Re z2(ω) forthe half-space coated by the piezo-film in figure 12 below.
The values of resonance semi-widths an , obtained forspecified ω by substituting zl from such marked curves into theexpression (4.1) instead of ξm, are also shown by vertical linesin subplots (d), (e) and (f) of figure 10. As can be seen, theymatch very well with the maxima of the total source energyE0(a). Hence, those values an may be referred to as points ofstrip width resonance at a fixed frequency.
In closing, we should remark that the principal branchcannot be selected in narrow bands in the vicinity of the cut-offfrequencies of new branches zl(ω) (e.g. for 2.7 < ω < 3.2).
4.2. Elastic half-space
Numerical analysis shows that energy radiation into the half-space substrate is also influenced by the strip width resonance(figure 11). With frequency increasing, the E0(a) plotsbecome very similar to those for the layer (e.g., comparesubplots (f) in figures 10 and 11). From the physical pointof view this is quite explainable by the growth of the relativethickness h/λ (λ is a wavelength) as ω increases, so that atω � 1 the layer behaves as a bottomless substrate. Onthe other hand, the wave structure in a half-space is quitedifferent from the normal mode superposition peculiar to alayer. There exists a single real Rayleigh pole ζR and a singleinterface guided wave in the contact zone specified by thereal zero z1 (figure 12). Besides, there are two body wavesalso contributing as Ev(a) in the source energy radiation toinfinity (see (2.32)). The infinite variety of evanescent waves
Figure 11. Like figure 10 but for a half-space substrate. In subplots(c)–(f) an = π(n + 1/2)/Re z2, where z2 is given in figure 12.
Figure 12. Dispersion curves in the case of a half-space hostmedium. Wavenumbers of body P and S waves (κ1 and κ2),Rayleigh waves (ζR), and surface waves of Rayleigh type in ahalf-space coated by a piezoceramic film (pure real z1 and complexz2). The circle is for the pure real value z2.
associated with complex zl , obtained from equation (2.24)for a patch–layer waveguide, degenerates with a half-spacesubstructure into a single decaying wave related to the onlycomplex root z2. Similar to z1, the dependence of Re z2 onω appears linear in figure 12 (almost dispersionless phasevelocity c2 = Re z2/ω), while the attenuation characteristicIm z2(ω) changes irregularly down to zero at a single pointnear ω ≈ 10.65 (the corresponding real value of z2 is markedin figure 12 by a circle).
The vertical lines in subplots 11(b) and (c) relate to theresonance values an calculated by replacing ξm by z1 in (4.1)and Re z2, respectively. The resonance maxima of the Rayleighpart of the source power ER(a) are well described at any ωby those an obtained using z1, while the values an calculatedthrough Re z2 approach the Ev(a) points of maxima as ωincreases. Here, the contribution of Ev to E0 becomes the mostimportant; therefore, at higher frequencies the an , calculatedthrough Re z2, determine the resonance maxima of E0(a) aswhole (subplots 11(d)–(f)).
The coincidence of these points with those obtained in thecontext of a layered model (figures 10(d)–(f)) is explained bythe fact that the principal sections of the branches in figure 3are aligned in a line near to Re z2(ω) in figure 12.
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Integral equation based modeling of patch–structure interaction
5. Selective mode excitation
The application of normal modes to ultrasonic inspection ofplate-like structures creates a need to work with a singletraveling mode over a limited frequency range [9]. Withtwo patch actuators bonded to opposite sides of a plate, thefundamental s0 and a0 modes may by excited separately byapplying the driving electric fields Ez,m in phase or out ofphase. However, in many practically important situations,one side of the inspected structure is inaccessible (e.g. pipes,tank walls, some parts of frames of chemical plants, engines,aircraft panels and so on). For this reason, simple phasing isnot sufficient for selectively exciting higher modes.
With a set of patch actuators bonded to one side of awaveguide only, the problem of exciting the required modesand their directional radiation may also be solved by a properselection of the driving fields Ez,m . The proposed, analyticallybased approach provides a possibility for strict calculation ofthe optimal driving parameters. A short communication aboutthe algorithm for selective mode excitation, developed in thecontext of a layer model, appeared in [17]. In the presentsection, we give a more detailed description, extending it to thecase of a half-plane substrate. The latter is not only for surfacetraveling waves, but also for body wave directional radiationinto the host medium.
5.1. Elastic layer
In the model considered, the generalized driving parametersare em = |em|eiθm (see equation (2.7)). With fixedgeometry, frequency and material properties, only the set ofem determines the contact stresses qm, which, being surfaceloads applied to the substructure, generate the wavefields ofgeneral form (2.18) or, specifically for the layer, the normalmode fields (2.22).
Let us separate the qm further, and represent them in theform
qm(x) =M∑
j=1
e j qm, j (x), m = 1, 2, . . . ,M. (5.1)
Here the qm, j are the solutions to the integro-differentialequations (2.19) with the boundary conditions
is Kronecker’s delta, and vm, j : vm = ∑j e jvm, j .
In other words, the functions
q( j) =∑
m
qm, j and v( j) =∑
m
vm, j (5.3)
are the contact stresses and patch displacements generated bya generalized unit electric field e j = 1 applied solely tothe j th actuator with zero voltage at all others (em ≡ 0 form �= j ). These functions are a kind of characteristic (Green’s)function for a piezoceramic transducer simulated by the patch–layer model under consideration. With any vector of driving
parameters e = {e1, e2, . . . , em}, the resulting characteristicscan be expressed in terms of the basis functions (5.3):
q =∑
j
e j q( j), v =
∑j
e jv( j), u =
∑j
e j u( j).
(5.4)It is noteworthy that the elementary functions qm, j and vm, j ,being solutions to the coupled problem with M strips, alsotake strictly into account the mutual strip influence due to waveinteraction through the substrate.
In view of q splitting (5.4), the related wavefields um
can be also expanded in terms of basis fields u( j)m excited by
the basis loads qm, j , just as the fields of each normal modein (2.22):
um =∑
j
e j u( j)m =
∑j
e j
∑k
u( j)m,k
u( j)m,k(x, z) = t±
m,k j a±k (z)e
±ζk (x−xm∓am), x /∈ �m
(5.5)
t±m,k j is of the same form as t±
m,k in (2.22) replacing Qm byQm, j = F[qm, j ]. Consequently, in (2.31), the energy of thekth mode E±
k is representable as the quadratic form
E±k =
M∑i
M∑j
ei e∗j (u
±k,i , u±
k, j ) = (A±k e, e) (5.6)
where u±k, j =
√ω
4rk
∑m t±
m,k j e∓iζk(xm±am) are combined from
the terms related to the first components of vectors u( j)m,k for
x > � and x < �, respectively. The A±k are the M × M
Hermitian matrices with elements ai j = (u±k, j , u±
k,i ) = a∗j i
(the inner product assumes complex conjugation of the secondfactor, the same as in (A±
k e, e)). Like the basis fields u( j),the matrices A±
k (ω) are independent of the driving electricfields. They define the possible kth mode excitation by thespecified transducer. The maximal and minimal values of theform (5.6) on a compact set of vectors e bounded in accordancewith some natural, practical limitations (e.g., |e j | < constor ‖e‖ = const) are obviously related to the maximum andminimum possible energies of the kth traveling wave radiatedto plus or minus infinity. Thus, the problem of exciting specificmodes is reduced to an optimization problem with an objectivefunction obtained from the quadratic forms (5.6).
For example, as the objective function we can choose
ν(e) = (Ae, e)/(Be, e) (5.7)
where A = ∑M+1
j=1 A+k j
+ ∑M−1
j=1 A−l j
and B = ∑M+2
j=1 A+m j
+∑M−2
j=1 A−n j
; k1, . . . , kM+1
and l1, . . . , lM−1
are the numbers ofmodes to be maximized in the right and left directions ofradiation, respectively, while m1, . . . ,m M+
2and n1, . . . , nM−
2
are the numbers of modes to be suppressed.The extremal points of ν(e) satisfy the extremum
conditions
∂ν(e)∂e j
= 0, j = 1, . . . ,M (5.8)
which leads to the following nonlinear system:
(A − νB)e = 0 (5.9)
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E Glushkov et al
with respect to the vector of unknown parameters e. If matrixB is non-singular (det B �= 0), the latter expression can berewritten as
Re − νe = 0, R = B−1 A, (5.10)
and the problem is reduced to searching for the eigenvectores of matrix R corresponding to its maximal eigenvalue λs :Res = λses . In such a case Aes = λs Bes , hence ν(es) = λs
and equation (5.10) holds identically.If matrix B is singular (det B = 0), then the eigenvector e0
associated with its eigenvalue λ0 = 0 being taken as e providesthe full damping of undesirable radiation: (Be0, e0) = 0. Inthe event of multiple eigenvectors e(i)0 associated with λ0 = 0,the vector es maximizing the form (Ae, e) is, obviously, to besearched for in the linear span of those vectors e(i)0 .
Let us derive sufficient conditions for making matrixB singular, i.e. the conditions for secure suppression ofundesirable modes. Since its elements
bi j =M+
2∑k=1
(u+mk , j , u+
mk ,i)+M−
2∑k=1
(u−nk , j , u−
nk ,i) ≡ (w j ,wi )
may be treated as inner products of vectors wi ={u+
m1,i, . . . , u+
mM+2,i , u−
n1,i, . . . , u−
nM−2,i}, i = 1, 2, . . . ,M , det B
is a gramian of M vectors wi of length M2 = M+2 + M−
2 . IfM > M2, the system of vectors wi is linearly dependent, hencedet B ≡ 0. If M � M2, then, in general, det B �= 0. In otherwords, M sources are enough to suppress completely N < Mtraveling modes (here, the modes u±
k going to plus and minusinfinity are counted as two different modes). All one has to dois to fix the driving parameters e meeting the condition Be = 0,i.e. to be an eigenvector associated with λ0 = 0.
For example, in the two-mode range 0 < ω < 3.14 asingle fundamental mode emitted only in one direction (e.g. tothe right of actuators) can be generated by four patches, whichis enough for suppressing the energy of the three unnecessarymodes: E−
1 = E−2 = E+
2 = 0 or E−1 = E−
2 = E+1 = 0
for radiation of the unique a0 wave mode u+1 or s0 mode u+
2 ,respectively. Plots of the surface horizontal displacementsu(x) = ux (x, 0) (both Re u and Im u) calculated with drivingvectors e providing such suppressions are depicted in subplots(a) and (b) of figure 13 (hereinafter, e is normalized so that|e| = 1).
These two examples are for ω = 2 and patches ofhalf-width am = 1. It is evident from the plots that thereis no backward radiation to the left, while to the right ofthe actuated zone � the displacements quickly attain a puresinusoidal form with spatial periods 2π/ζ1 (figure 13(a)) and2π/ζ2 (figure 13(b)). The third subplot 13(c) illustrates thatit is possible to generate a so-called backward mode, definedby opposite phase and group velocities (u5 with cp,5 < 0 andcg,5 > 0 at ω = 5.5 in this case). Since there may exist fivemodes associated with poles ζk at this frequency (figure 3), fiveactuators are required to suppress four of them to the right sideof actuators, while the left direction does not participate in thisexample.
5.2. Elastic half-space
Since surface Rayleigh waves excited in a half-space arerepresented in form (2.27) (first line), in a similar way to
Figure 13. Normal mode selective excitation in a layer. Singlefundamental modes a0 and s0 at ω = 2 (a) and (b), and backwardmode ζ5 propagating to the right at ω = 5.5 (c).
the Lamb modes (2.22), controlling their radiation is just aspecial case of the algorithm described above. Controllingbody wave radiation ((2.27), third line) leads to a slightlydifferent optimization problem. Though the objective functionν(e) has the same general form (5.7) (it has been introducedin [18] for a similar optimization problem but for a group ofseismic vibrators), the specific representation of matrices Aand B is different. In addition, the main interest in work withbody waves is the control of radiation directivity.
Let the required direction be an angular sector ϕ1 � ϕ �ϕ2 in polar coordinates x = R cosϕ, z = R sinϕ, R =√
x2 + z2, −π < ϕ < 0, while in the remaining directions−π < ϕ < ϕ1 and ϕ2 < ϕ < 0 the radiation has to beminimized. It seems natural to introduce the goal function as aratio of the energy flux Ev,1 through the part of the cylindricalsurface ϕ1 < ϕ < ϕ2, R � 1, to the energy flux Ev,2 throughthe rest of the surface:
ν(e) = Ev,1/Ev,2. (5.11)
In accordance with the general representation (2.28)
Ev,1 =∫ ϕ2
ϕ1
ev(R, ϕ)R dϕ as R → ∞ (5.12)
where the power density ev is calculated by retaining the mainterms of the body wave asymptotics (2.27) only. Ev,2 is of thesame form except for the interval of integration ([−π, ϕ1] ∪[ϕ2, 0] instead of [ϕ1, ϕ2]). In a homogeneous medium ev splitsinto the P and S wave power density:
ev = ep + es, ep = ω
2κ1(λ+ 2μ)|u p|2,
es = ω
2κ2μ|us|2
(5.13)
where u p and us are amplitudes of the first (n = 1) and second(n = 2) terms of the body wave asymptotics (2.27) for the
662
Integral equation based modeling of patch–structure interaction
total field∑
m um. In fact, they are displacement projectionson the radial and transversal directions np = {cosϕ, sinϕ} andns = {− sinϕ, cos ϕ}, respectively.
Much as above, the body wave asymptotics also split upin terms of basis fields u( j) related to unit driving parametersem = δm j . This paves the way to a reduction of the goalfunction (5.11) to a ratio of quadratic forms Ev,1 = (Ae, e)and Ev,2 = (Be, e). Specifically, these representations followfrom the expressions for displacement amplitudes in (5.13):
|u p(ϕ)|2 =∣∣∣∣∑
j
e j t(1)j (ϕ)
∣∣∣∣2/
R,
|us(ϕ)|2 =∣∣∣∣∑
j
e j t(2)j (ϕ)
∣∣∣∣2/
R
(5.14)
where t(n)j = ∑m t(n)j,m , n = 1, 2. The t(n)j,m are of the same form
as t(n)m in (2.27) except that Qm is replaced by Qm, j .The elements ai j and bi j of matrices A and B are
obtained by integration of the corresponding expressions overthe respective intervals ϕ. Here,
ai j = a pi j + as
i j
a pi j = ω
2(λ+ 2μ)κ1
∫ ϕ2
ϕ1
(t(1)i (ϕ), t(1)j (ϕ)) dϕ
and so on.Often, it is necessary to control only one kind of waves
(either P or S). In such cases, only the corresponding terms(a p
i j or asi j ) are retained in the goal functions. The remaining
calculations (searching for the maximizing eigenvectors es) areas before.
We should remark that body waves result from continuousspectra, 0 � α � κn. Therefore, unlike normal modes,which are associated with the discrete spectrum of the problem(poles ζk), they cannot be suppressed completely when a finitenumber of actuators are used.
The ability to focus body waves in a given directionis illustrated by the polar patterns in figure 14. Theydepict the amplitude factors |u p|
√R and |us|
√R defined in
equation (5.14) versus polar angle ϕ for P (left subplots (a))and S (right subplots (b)) waves generated by ten actuators ofsize am = 1 spaced uniformly on the surface from x1 = −14to x10 = 14. The angular domains [ϕ1, ϕ2], in which theradiation is maximized, are marked by bold arcs on the lowersemi-circles. It can be seen in these diagrams that the proposedmethod allows for the possibility of directional illumination ofthe substructure by elastic body waves.
6. Concluding remarks
Our theoretical model for elastic waves excited by flexiblepiezoelectric patch actuators is intended, first of all, for thedevelopment of structural health monitoring systems. Ourmodel, based on an integral equation approach, providesquantitative solutions in a wide (multi-mode) frequency band,like FEM simulations, rigorously accounting for the patch–structure contact interaction. In comparison to FEM solutions,however, it is much less time consuming, and gives insights
Figure 14. Directional radiation of P and S body waves (left andright subplots (a) and (b)), respectively, optimized for the angulardomains shown by bold arcs.
into the wave structure, by providing solutions in terms ofguided and body waves. In this respect the new model issimilar to conventional, simplified engineering approaches, butextends considerably their range of application. Its utilityin the investigation of fine wave phenomena is illustratedby examples of patch–layer resonance effects, wave energypartition, selective mode excitation and directional radiation.
Though the method is presented for the 2D case, it isin general quite applicable to 3D problems as well. Theextension to 3D requires much less modification than onemight expect. First, in outward appearance, the coupledintegro-differential problem (2.19) remains as in 2D, althoughthe scalar operators K and M become matrix ones withrespect to unknown vectors of shear contact stress q(x, y)and displacement v(x, y), (x, y) ∈ � = ∪m�m. Numericalsolutions to these equations can be obtained by expansion interms of axially symmetric δ-like basis functions. This methodhas proved its efficiency with similar integral equations in 3Ddynamic contact problems [18, 20], and in 3D elastodynamicscattering by arbitrarily shaped cracks [25].
Acknowledgments
We are thankful to Professor R Williams, the University ofTennessee, for helpful discussions on wave phenomena and forimproving the English of this paper. This work is supported bya grant of the INTAS co-operative project No 05-100008-7979and by Russian Foundation for Basic Research (RFBR) grants.
References
[1] Crawley E F and de Luis J 1987 Use of piezoelectric actuatorsas elements of intelligent structures AIAA J. 25 1373–85
[2] Chaundhry Z and Rogers C A 1994 The pin-force modelrevisited J. Intell. Mater. Syst. Struct. 5 347–54
[3] Banks H T, Smith R C and Wang Y 1995 The modelling ofpiezoceramic patch interactions with shells, plates andbeams Q. Appl. Math. 53 353–81
[4] Seemann W 1996 A linear ultrasonic traveling wave motor ofthe ring type Smart Mater. Struct. 5 361–8
[5] Giurgiutiu V and Zagrai A N 2000 Characterization ofpiezoelectric wafer active sensors J. Intell. Mater. Syst.Struct. 11 959–75
[6] Giurgiutiu V 2003 Lamb wave generation with piezoelectricwafer active sensors for structural health monitoring Proc.SPIE 5056 111–22
[7] Raghavan A and Cesnik C E S 2005 Finite-dimensionalpiezoelectric transducer modeling for guided wave basedstructural health monitoring Smart Mater. Struct.14 1448–61
[8] Liang C, Sun F P and Rogers C A 1994 An impedance methodfor dynamic analysis of active material systems J. Vib.Acoust. (Trans. ASME) 116 120–8
[9] Wilcox P D, Lowe M J S and Cawley P 2001 Mode andtransducer selection for long range wave inspection J. Intell.Mater. Syst. Struct. 12 553–65
[10] Liu G R 2002 A combined finite element/strip element methodfor analyzing elastic wave scattering by cracks andinclusions in laminates Comput. Mech. 28 76–81
[11] Moulin E, Assaad J and Delebarre C 2000 Modeling of Lambwaves generated by integrated transducers in compositeplates using a coupled finite element-normal modesexpansion method J. Acoust. Soc. Am. 107 87–94
[12] Seemann W 1997 Stresses in a thin piezoelectric elementbonded to a half-space Appl. Mech. Rev. 50 204–9
[13] Zhang B, Bostrom A and Niklasson A J 2004 Antiplane shearwaves from a piezoelectric strip actuator: exact versuseffective boundary condition solutions Smart Mater. Struct.13 161–8
[14] Wang X D and Huang G L 2001 Wave propagation inelectromechanical structures: induced by surface-bondedpiezoelectric actuators J. Intell. Mater. Syst. Struct. 12105–15
[15] Kochetkov I D and Rogacheva N N 2005 Contact interaction ofa piezoelectric actuator and elastic half-space J. Appl. Math.Mech. 69 792–804
[16] Glushkov E V, Glushkova N V, Seemann W andKvasha O V 2006 Elastic wave excitation in a layer bypiezoceramic patch actuators Acoust. Phys. 52 398–407
[17] Glushkov E V, Glushkova N V, Kvasha O V andSeemann W 2006 Selective mode excitation in elasticwaveguides by piezoceramic actuators AIP Conf. Proc.834 152–9
[18] Babeshko V A, Glushkov E V and Zinchenko J F 1989 TheDynamics of Inhomogeneous Linearly-Elastic Bodies(Moscow: Nauka) (in Russian)
[19] Glushkov E V and Glushkova N V 1997 Blocking property ofenergy vortices in elastic waveguides J. Acoust. Soc. Am.102 1356–60
[20] Glushkov E V and Glushkova N V 2001 On the efficientimplementation of the integral equation method inelastodynamics J. Comput. Acoust. 9 889–98
[21] Glushkov E, Glushkova N, Golub M and Bostrom A 2006Natural resonance frequencies, wave blocking, and energylocalization in an elastic half-space and waveguide with acrack J. Acoust. Soc. Am. 119 3589–98
[22] Aki K and Richards P G 1980 Quantitative Seismology: Theoryand Methods (New York: Freeman)
[23] Glushkov E V, Glushkova N V and Kvasha O V 2005 Solutionof a dynamic contact problem for an elastic layer withpiezoceramic surface-bonded patches Available from VINITINo. 1015-B2005 Krasnodar Kuban State University, pp 1–40(in Russian)
[24] Seemann W, Ekhlakov A, Glushkov E V, Glushkova N V andKvasha O V 2007 The modeling of piezoelectrically excitedwaves in beams and layered substructures J. Sound Vib.301 1007–22
[25] Glushkov E, Glushkova N, Ekhlakov A and Shapar E 2006An analytically based computer model for surfacemeasurements in ultrasonic crack detection Wave Motion43 458–73
