DOI: 10.1177/1045389X12467516 and propagation in plates with … · 2019. 5. 29. · modeling of guided wave excitation and propagation in plates with applied prestresses F. Song1,

Article

Journal of Intelligent Material Systemsand Structures24(5) 598–611� The Author(s) 2012Reprints and permissions:sagepub.co.uk/journalsPermissions.navDOI: 10.1177/1045389X12467516jim.sagepub.com

Coupled piezo-elastodynamicmodeling of guided wave excitationand propagation in plates with appliedprestresses

F. Song1, G. L. Huang1 and G. K. Hu2

AbstractPlate-like aerospace engineering structures are prone to mechanical/residual preloads during flight operation. This articlefocuses on the quantitative characterization of applied prestress effects on the piezoelectrically-induced guided wavepropagation, which has been widely used in structural health monitoring systems. An analytical model consideringcoupled piezo-elastodynamics is developed to study dynamic load transfer between a surface-bonded thin piezoelectricactuator and a prestressed plate. The accuracy of the analytical prediction is evaluated by the comparison with the finiteelement analysis. Based on the developed model, the load-dependent guided wave signal variation in both time-of-flightand amplitude is determined, and its dependence on loading frequency and host material properties is also discussed. Itis found that the guided wave signal variation due to the prestress could be significant under some circumstances. A sig-nal difference coefficient is finally proposed to quantitatively assess the signal variation caused by different prestresses.This study can serve as a theoretical foundation for the development of the real-time piezo-guided wave–based struc-tural health monitoring system in a realistic loading environment.

KeywordsPiezo-elastodynamic modeling, dynamic load transfer, guided wave generation, prestress/residual stress, guided wave–based structural health monitoring

Introduction

Ultrasonic guided waves (GWs), which can propagateover large distances in plates and shells, have showngreat potential for structural health monitoring (SHM)of thin-walled aircraft and aerospace structures overthe past decades (Anton et al., 2009; Kundu andMaslov, 1997; Senesi and Ruzzene, 2011; Song et al.,2012). Due to advantages such as quick response, lowpower consumption, low cost, and small size, piezoelec-tric wafers are particularly suitable for being integratedin those structures as actuators/sensors to form anactive on-line SHM system (Giurgiutiu, 2005; Ihn andChang, 2008; Park et al., 2010; Raghavan and Cesnik,2007; Song et al., 2009). Many damage diagnosis andcharacterization approaches have been developed andsuggested by comparing the GW signals from the dam-aged structures to the baselines recorded from theundamaged structures (Huang et al., 2010b; Lin andYuan, 2005; Michaels, 2008). However, implementa-tion of those approaches are mostly based on the con-dition that the monitored structures are under the ideal

laboratory environment, in which the working loads onthe monitored structures such as mechanical/residualprestresses are not considered.

It is well known that operating aircraft and aero-space structures are often subjected to various pres-tresses such as applied loads. During manufacture andassembly, many load-bearing elements of aircraft suchas fuselages, longerons, stringers, bulkheads, and stabi-lizers are fully or partially prestressed to enhance theirfatigue strength and fracture resistance (Ratwani,2000). In bolted aerospace structures, applied stressescan be exerted by the bolts on the surrounding materialof the joints (Doyle et al., 2010). Significant thermal

1Department of Systems Engineering, University of Arkansas at Little

Rock, Little Rock, AR, USA2School of Aerospace Engineering, Beijing Institute of Technology, Beijing,

China

Corresponding author:

G. L. Huang, Department of Systems Engineering, University of Arkansas

at Little Rock, 2801 S University Avenue, Little Rock, AR 72204, USA.

Email: [email protected]

residual stresses in aerospace composite structures canbe initiated due to the difference in thermal expansioncoefficient (CTE) and mechanical incompatibility ofthe different components when the structures serveunder extreme temperature environment (Karamiet al., 2008). Thin-walled aerospace structures are usu-ally operated under in-plane loads during flight service.For instance, spinning helicopter rotor blades, bladeddisks in engines, and pressurized aircraft cabins areunder tensile loads while buoyant structures are undercompressive loads due to gravitational forces or pres-sure loads (Lesieutre, 2009). Therefore, considerationof prestress effects on the GW excitation and propaga-tion in thin-walled aerospace structures is of practicalimportance in the design of an in situ SHM system.

A number of research activities have been conductedon characterization of mechanical and dynamic beha-viors of various prestressed structures (Akbarov, 2007;Hu et al., 2002; Kwun et al., 1998; Loveday, 2009;Singh, 2010; Tiersten et al., 1981; Yang, 2005) by usinganalytical, numerical, or experimental methods. Desmetet al. (1996) studied effects of the externally appliedstress on the fundamental Lamb wave propagation inpolymer foils. It was found that the applied tensilestress had very small influence on the dispersion of thesymmetric mode compared to the antisymmetric mode.Based on the finite element modeling, Chen and Wilcox(2007) examined pretension influences on the GW pro-pagation in cables and rails, and found that the wavevelocity of the fundamental flexural wave changed sig-nificantly due to the appearance of tensile load, espe-cially for low-frequency cases. An and Sohn (2010)tested the transient GW responses in plates under vari-ous static loading conditions and similar conclusionswere drawn. Doyle et al. (2010) and Zagrai et al. (2010)experimentally studied the acoustoelastic effects ofpiezoelectric wafer-active sensors (PWAS) bonded ontothe bolted space structures under different prestresses.Recently, the effects of applied loads on GW responseswere experimentally investigated in the context of envi-ronmental load impacts on GW-based SHM usingpiezoelectric wafers (Michaels et al., 2011). To the bestof our knowledge, however, no systematic theoreticalstudy has been carried out to quantitatively characterizeprestress effects on the coupled piezo-elastodynamicbehavior of surface-bonded piezoelectric wafers. Suchtheoretical framework would be fundamental in under-standing the reliability and feasibility of applying piezo-GW–based SHM systems for field applications.

Based on the Bernoulli–Euler beam theory, Crawleyand De Luis (1987) developed a static shear-lag modelto study the interfacial load transfer between a piezo-actuator and a substructural coupled beam for low-frequency cases. A direct extension of the classicalshear-lag theory (Crawley and De Luis, 1987) has beenreported (Giurgiutiu and Santoni, 2009), where thenonlinear stress distribution across the thickness was

considered. To avoid the difficulties associated with thecomplex interaction between the actuator and the hostmedium, the interfacial shear stress was simplified as‘‘pinching’’ tangential traction only at the actuator tipson the structure surface by neglecting coupled actuator/structural dynamics (Raghavan and Cesnik, 2005; VonEnde and Lammering, 2007). To consider the coupledpiezo-elastodynamic behavior, a one-dimensionalactuator model has been developed to investigate thecoupled dynamic load transfer between the piezo-actuator and the semi-infinite elastic medium (Huang etal., 2010b; Huang and Sun, 2006; Wang and Huang,2001). Similarly, an integral equation-based model wasalso proposed to study the patch–layer interaction andresonance effects in a system of flexible piezoelectricpatch actuators bonded to an elastic substrate(Glushkov et al., 2007). However, the piezo-actuatormodel surface bonded to a prestressed plate structurehas not been developed.

In this article, we extend the one-dimensional actua-tor model (Huang and Sun, 2006; Wang and Huang,2001) to quantitatively investigate the interfacial shearstress and its resulting GW propagation in the pre-stressed isotropic plate. The interfacial shear stressbetween the actuator and the host plate is calculatedusing the Fourier transform technique and solving theresulting integral equations. The analytical predictionsof the interfacial shear stress and transient responsesfrom the current model are compared with the finiteelement simulation results to evaluate the accuracy ofthe developed model. Based on the current model andthe modified Rayleigh–Lamb equations in the pre-stressed plate, the load-dependent GW excitation andpropagation by the piezoelectric wafer actuator is stud-ied, from which the prestress effects are demonstrated.It is found that the wave response variation induced bythe prestresses could be significant under some circum-stances, which is quantitatively characterized by a pro-posed signal difference coefficient (SDC).

Formulation of the problem

Consider a plane strain problem of a thin piezo-actuator surface bonded to an isotropic host plate sub-jected to a uniform prestress, as sketched in Figure 1.In this figure, the half-length and the thickness of theactuator are denoted as a and h, respectively, and thehalf-thickness of the host plate is d. The host plate isunder a uniaxial static prestress s0 along the x direc-tion. The poling direction of the actuator is along thez-axis, and a voltage (DV ) between the upper and lowerelectrodes of the actuator is then applied, which resultsin an electrical filed (Ez) of circular frequency v alongthe poling direction of the actuator. For the steady-state harmonic solution of the system, the time factorexp �iv tð Þ, which applies to all the field variables, is

Song et al. 599

suppressed in the following analysis. In the currentmodel, it is assumed that the piezo-actuator is bondedonto the substructure after manufacturing.Conceptually, this is an ideal assumption for theachievement of online SHM based on surface-bondedpiezo-wafers (Giurgiutiu, 2008). Under this conceptualassumption, our model is mainly for considering theeffects of applied prestresses (during operation) onpiezo-actuated GWs. The perfect actuator/structurebonding is assumed in the current model and the adhe-sive interface layer effects (Dugnani, 2009; Han et al.,2009; Jin and Wang, 2011; Lanza di Scalea andSalamone, 2008) are neglected.

Modeling of the prestressed piezoelectric actuator

In this study, attention is paid to modeling the thinactuator, for which the thickness is very small com-pared with its length. In response to an electric field(Ez) applied across its thickness along the poling direc-tion, the actuator mainly experiences axial deformation.Therefore, the prestressed actuator can be modeled asan electroelastic element subject to the applied electricfield and the distributed interfacial stress, as illustratedin Figure 2. In this figure, ŝ0 is the resulting initial stressin the actuator due to the prestress s0 on the host plate,and t represents the interfacial shear stress transferredbetween the actuator and the host plate.

It is assumed that the incremental stress sax and dis-placement uax are uniform across the actuator. Underplane strain deformation, the axial incremental stress inthe actuator (sax) can be expressed as

sax =Eaeax � eaEz ð1Þ

where Ea and ea are the effective elastic and piezoelec-tric material constants, Ea = c11 � c213

�c33 and

ea = e13 � e33c13=c33, respectively, and Ez = � DV=hwith cij and eij(i, j = 1, 3, or x, z) being the componentsof the piezoelectric elastic stiffness matrix for a constantelectric potential, and the piezoelectric constant matrix,respectively. The small strain component is given by

eax =duaxdx

ð2Þ

By considering the incremental deformation (Su etal., 2005) together with equations (1) and (2), the equa-tion of motion of the prestressed actuator can be writ-ten as

Ea + ŝ0ave

� � d2uaxdx2

+t xð Þ

h+ rav

2uax = 0 ð3Þ

where ra is the mass density of the actuator, andŝ0ave = 1=2a

Ð 2a0

ŝ0 xð Þdx is an averaged resulting stressover the actuator length for simplification of the prob-lem. The determination of the resulting initial stressŝ0 xð Þ can be found in Appendix 1. Since the load trans-ferred between the actuator and the host plate can beattributed to t, two ends of the actuator during incre-mental deformation can be assumed as

sax = 0, xj j= a ð4Þ

The axial strain of the actuator can be obtained interms of the interfacial shear stress t by solving equa-tion (3) as

eax = eE xð Þ+sin ka a+ xð Þ

h Ea + ŝ0ave� �

sin 2kaa

ða�a

cos ka z � að Þt zð Þdz

�ðy�a

cos ka z � xð Þt zð Þ

h Ea + ŝ0ave� �dz

ð5Þ

where eE xð Þ= eaEz�

Ea + ŝ0ave

� �� cos kax=cos kaað Þ,

ka =v=ca, and ca = Ea + ŝ0ave

� ��ra

� �1=2with ka and ca

being the wave number and the axial wave velocity ofthe prestressed actuator, respectively.

GW solution of the prestressed plate structure

Under the plane strain deformation, the elastic wavemotion for a prestressed material can be expressed asfollows (Akbarov, 2007; Liu et al., 2003; Wang et al.,2007)

r2u+ s0

l+ 2m

� �∂2u∂x2

+v2

c2lu= 0 ð6Þ

Figure 1. An actuator surface-bonded to an elastic plate under the static prestress.

600 Journal of Intelligent Material Systems and Structures 24(5)

r2c+ s0

m

� �∂2c

∂ x2+

v2

c2tc= 0 ð7Þ

where r is the mass density of the host medium,c2l = l+ 2mð Þ=r, and c2t =m=r with l and m being theLame constants, respectively, and u and c are the two dis-placement potentials that satisfy the following equation

ux =∂u∂x

+∂c

∂zand uz =

∂u∂z� ∂c

∂xð8Þ

The general solution of the wave propagation can bedetermined by solving equations (6) and (7) using thespatial Fourier transform as follows

�f jð Þ=ð+‘�‘

f xð Þ eij xdx and f xð Þ= 12p

ð+‘�‘

�f jð Þ e�ij xdj

ð9Þ

The solutions can be expressed as follows

�u=A1 sin pzð Þ+A2 cos pzð Þ ð10Þ�c=B1 sin qzð Þ+B2 cos qzð Þ ð11Þ

where p2 = v2�

c2l� �

� 1+ s0�

l+ 2mð Þ� ��

j2 andq2 = v2

�c2l

� �� 1+ s0

�m

� �� j2. Using the linear

strain–displacement relation and the constitutive equa-tion, the stresses can be readily determined as

�szz =mj2 � q2� �

sin pzð ÞA1 + j2 � q2� �

cos pzð ÞA2�2jq cos qzð Þ iB1 + 2jq sin qzð Þ iB2

ð12Þ

�sxz =m2jp cos pzð Þ iA1 � 2jp sin pzð Þ iA2+ j2 � q2� �

sin qzð ÞB1 + j2 � q2� �

cos qzð ÞB2

ð13Þ

Based on the present actuator model, the boundarycondition on the upper surface of the host plate can bewritten as

sxz x, dð Þ= � t xð Þ, xj j\a and szz x, dð Þ= 0 ð14Þ

Using equations (12)–(14) and applying the inverseFourier transform, the Lamb strain component ex on

the surface z= d can be expressed in terms of the inter-facial shear stress t as follows

ex x, dð Þ= �1

4pm

ð+‘�‘

t hð Þð+‘�‘

iNS jð ÞDS jð Þ

+NA jð ÞDA jð Þ

e�ij x�hð Þdjdh

ð15Þin which NS jð Þ= jq cos pdð Þ cos qdð Þ j2 + q2

� �;DS jð Þ

= j2 � q2� �2

cos pdð Þ sin qdð Þ+ 4j2pq sin pdð Þ cos qdð Þ;NA jð Þ= � jq sinðpdÞ sinðqdÞ ðj2+q2Þ; and DAðjÞ=ðj2

�q2Þ2 sin ðpdÞ cos ðqdÞ+ 4j2pq cos ðpdÞ sin ðqdÞThe integral in equation (15) is singular at the roots

of Ds and DA, which represent the symmetric andantisymmetric eigenvalues of the modified Rayleigh–Lamb equations at frequency v with the prestressparameter s0. At low frequencies, only two eigenva-lues jS0 and j

A0 exist, while several eigenvalues appear

at sufficiently high frequencies. For s0 = 0, the dis-persion equations can be reduced to the conventionalRayleigh–Lamb equations without load (Raghavanand Cesnik, 2005).

Dynamic load transfer and resulting GW propagation

The continuity between the actuator and the host plateat z= d can be described as

eax xð Þ= ex x, dð Þ, xj j\a ð16Þ

By substituting equations (5) and (15) into equation(16), the integral equation can be obtained as

� 14pm

ð+‘�‘

t hð Þð+‘�‘

iNS jð ÞDS jð Þ

+NA jð ÞDA jð Þ

e�ij x�hð Þdjdh

� sin ka a+ xð Þh Ea + ŝ0ave� �

sin 2kaa

ða�a

cos ka z � að Þt zð Þdz

+

ðx�a

cos ka z � xð Þt zð Þ

h Ea + ŝ0ave� �dz = eaEz

Ea + ŝ0ave� � cos kax

cos kaa

ð17Þ

Figure 2. The schematic diagram of the actuator model.

Song et al. 601

Equation (17) is a singular integral equation of the firstkind, which involves a square-root singularity of t at theends of the actuator. The general solution of t can beexpressed in terms of Chebyshev polynomials, such that

t xð Þ=X+‘j= 0

cjTj x=að Þ� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� x=að Þ2q

ð18Þ

with Tj being Chebyshev polynomials of the first kind.If the expression in equation (18) is truncated to the

N-th term and equation (17) is satisfied at the followingcollocation points along the length of the actuator

xl = a cosl � 1N � 1 p� �

, l = 1, 2, . . . ,N ð19Þ

N linear algebraic equations in terms of cf g = c1,fc2, . . . , cN�1gT can be obtained as

M½ � cf g= Ff g ð20Þ

where M½ � is a known matrix given by

Mlj = �n02m

X+‘j= 1

cjsin½jcos�1�xl�sin½cos�1�xl�

+n02m

X+‘j= 1

cj

ð+‘0

Plj�j,�xl� � NS �j� �

n0DS �j� � + NA �j

� �n0DA �j

� � + 1" #

d�j

+a

h Ea + ŝ0ave� �X+‘

j= 1

cj

ðpcos�1xl

cos �ka cos u� �xl� ��

cos juð Þ du

+ap

h Ea + ŝ0ave� � sin �ka �xl + 1� ��

sin 2�ka� � X+‘

j= 1

cjP2j

with n0 = 2 1� nð Þ (n is the Poisson’s ratio of the hostmedium),

�xl = xl�

a, �ka = �kaa, �j= ja,

Plj�j,�xl� �

= Jj �j� � �1ð Þn cos �j�xl� � j= 2n+ 1

�1ð Þn+ 1 sin �j�xl� �

j= 2n

(,

P2j = Jj�ka� � �1ð Þn sin �ka� � j= 2n+ 1

�1ð Þn cos �ka� �

j= 2n

(

with Jj (j = 1, 2,.) being the Bessel functions of thefirst kind, and Ff g is the applied load withFl = eE xl

� �, l = 1, 2,., N. From these equations, the

unknown coefficients in cf g can be determined, fromwhich the interfacial shear stress t can be obtained.By applying the residue theorem (Raghavan andCesnik, 2005) and using the resulting cf g in equations(18) and (15), the steady-state Lamb strain inducedby the piezo-actuator on the surface z= d can bedetermined as

ex x, dð Þ=�pi2m

XNj= 1

cj

�1ð Þn

P�j

S

NS �jSð Þ

D9S �jSð ÞJj

�jS

�cos �j

S�xl

�

+P�j

A

NA �jAð Þ

D9A �jAð ÞJj

�jA

�cos �j

A�xl

�26664

37775, j= 2n+ 1

�1ð Þn+ 1

P�j

S

NS �jSð Þ

D9S �jSð ÞJj

�jS

�sin �j

S�xl

�

+P�j

A

NA �jAð Þ

D9A �jAð ÞJj

�jA

�sin �j

A�xl

�26664

37775, j= 2n

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð21Þ

in which �jS = jS�

a and �jA = jA�

a. It should be men-tioned that while a single circular frequency v isassumed for harmonic excitation in the above analysis,the transient GW response to any time-limited signalcan be evaluated by taking the inverse Fourier trans-form of the integral of the product of the harmonicresponse multiplied by the Fourier transform of theexcitation signal over the bandwidth (Raghavan andCesnik, 2005).

Validation of the model

To evaluate the developed model, predictions from thecurrent model will be compared with the results byusing commercially available finite element (FE) codeANSYS/Multiphysics 11.0. The two-dimensionalPLANE13 element with four nodes and three degrees-of-freedom (DOF) at each node is selected for thepiezo-actuator and the plate. The additional DOF inthe coupled field element is the electrical voltage. Theinput voltage can be applied on the top nodes ofthe piezo-actuator, and zero voltage is assigned for allthe bottom nodes of the piezo-actuator to simulate thegrounding operation. The material properties of theactuator used for the following numerical examples aregiven in Table 1.

Static deformation analysis is first conducted tocompute the initial interfacial stress distributionbetween the piezo-actuator and the host plate causedby prestresses. Transient analysis is then performed tocalculate the dynamic response of the prestressed plate(Chen and Wilcox, 2007).

Initial interfacial load transfer

First, consider the initial static deformation of thepiezo-actuator surface bonded to the prestressed plate.The material properties of the actuator can be found inTable 1, and E = 2:7431010 Pa and n= 0:3 are usedfor the host plate (Wang and Huang, 2001). Figure 3


shows the normalized initial static interfacial shearstress distribution between the actuator and the pre-stressed host plate with varying actuator half-length-to-thickness ratios (a=h). In this figure, t0�= t

�s0

�� ,ra=r= 1, and H=h= 40 with H = 2d being the platethickness. In this analysis, 40 terms in Chebyshev poly-nomial expansions are used. It can be seen that theinterfacial shear stress more and more concentrates theactuator tips with the increase of the actuator length-to-thickness ratio, as predicted by both the currentmodel and the FE simulation. Excellent agreementbetween the current model and the FE simulation isobserved especially for the actuator with large length-to-thickness ratios, that is, a=h.10 that validates theapplicability of the current one-dimensional actuatormodel and the averaged resulting stress assumption inthe integrated piezo-plate structure.

Transient GW propagation

To further show capability of the developed model, thetransient GW signals generated by the piezo-actuatorsurface bonded to prestressed plates are evaluated inthis subsection. The excitation used in the study is afive-peak tone-burst signal (Song et al., 2012), whichhas been widely used for the GW-based SHM. Toobtain the theoretical transient GW response, theinverse Fourier transform of the harmonic solution isconducted over the frequency spectrum of the excitedtone-burst signal. Figure 4 shows the normalized tran-sient strain responses predicted by the current model(equation (21)), and the FE model at a central excita-tion frequency of fc = 100 kHz. The material propertiesof the actuator are given in Table 1 and the host platematerial is assumed to be a typical aircraft-grade alumi-num alloy (E = 79 GPa, y = 0:33 and r= 2600 kgm�3)with the yield strength around sY = 0:5 GPa (Santusand Tayler, 2009), respectively. In Figure 4, Am� is thenormalized surface strain component ex with respect toits maximum value. In the example, the applied pres-tress is s0 = 0:8sY , geometry parameters are a=h= 20,H=h= 13:3 with H = 2d, and the strain is calculated atx = 226.5 mm away from the actuator. In Figure 4, acomplete separation of the fundamental symmetricmode (S0) and antisymmetric mode (A0) can beobserved. Specifically, very good agreement in bothwave amplitude and phase can be clearly seen betweenthe current model and the FE simulation, which showsthe capability of the current model in quantitativelydescribing the GW excitation and propagation in theprestressed plate.

Effects of prestresses on GW propagation

In this section, attention is paid to characterization ofthe prestress effects on the GW generation and propa-gation based on the developed analytical model. Inthe following calculation, the actuator materialproperties are listed in Table 1, and the typical aircraft-grade aluminum alloy (E= 79 GPa, y = 0:33 andr= 2600 kgm�3) with the yield strength aroundsY = 0:5 GPa (Santus and Tayler, 2009) is consideredas the host plate. The static prestresses varying in therange of 60.4 GPa are considered, where the positiveand negative prestresses physically denote tension andcompression, respectively.

Dispersion of the fundamental Lamb wave modes

Figure 5 shows the dispersion curves of the fundamen-tal S0 and A0 Lamb wave modes in the plate under dif-ferent prestresses. In this figure, cp and ct represent thephase velocity of Lamb waves and the transverse bulkwave velocity, respectively, and f–H is the product offrequency and plate thickness. In Figure 5(a), the dis-persion properties of the S0 mode show small sensitivityto the applied prestresses throughout all considered fre-quencies. For instance, the predicted change in the velo-city is less than 0.25% for most frequencies even whenthe stress is applied up to 0:8sY . However, for the A0mode in Figure 5(b), an obvious shift in the wave velo-city can be seen with the change of the applied pres-tresses especially at relatively low frequencies. It can befound that phase velocities of the A0 mode increasewith the increase of tensile stresses and decrease withthe increase of compressive stresses. The A0 mode hasbeen widely used to detect various damages in metallicplates due to its high sensitivity to structural damage(Francis Rose and Wang, 2010; Fromme and Sayir,2002). Therefore, when the A0 mode is used to evaluatethe structures under prestressed work conditions, thewave signal variation caused by the prestress should betaken into account. Interestingly, stopping bands of theA0 mode are observed in compressed plates for the rela-tively low frequencies, which represent evanescent wavepropagation.

Figure 6 shows the velocity variation of GW modesin the prestressed plate with different material proper-ties. In this figure, the normalized prestress-inducedwave velocity variation is defined as

Dc� s0� �

= cp s0

� �� cp s0 = 0

� �� cp s

0 = 0� �

ð22Þ

Table 1. Material properties of the piezo-actuator (Wang and Huang, 2001).

c11 (1010 N/m2) c13 (10

10 N/m2) c33 (1010 N/m2) e31 (C/m

2) e33 (C/m2) ra (kg/m

3)

13.9 7.43 11.5 25.2 15.1 7500

Song et al. 603

where s0 = 0:8sY , and the normalized Young’s modu-lus of the host plate is E�=E=E0 with E0 = 79 GPa.The excitation frequency is selected to be fc = 150 kHz.As observed in Figure 6, the softer the host structuralmaterial is, the more prestress-induced variation in thewave velocity can be found for both GW modes. Forexample, the effects of the prestress on the GW-basedSHM may become more profound for the aerospace

structures made of aluminum alloys (E�= 1:0) thanthose made of high-strength steels (E�.1:6).

Tuned GW generation

The technique of wave frequency tuning, which cangenerate the strong desired GW mode sensitive to spe-cific damages, has been widely used for GW inspection

Figure 4. The transient guided wave response in the prestressed plate at x = 226.5 mm away from the piezo-actuator underexcitation fc = 100 kHz (s

0 = 0:8sY ).FE: finite element.

Figure 3. The normalized initial static interfacial shear stress of the piezo-actuator surface-bonded to the prestressed plate withvarying actuator half-length-to-thickness ratio a/h.FE: finite element.


of plate structures (Gangadharan et al., 2009;Giurgiutiu, 2005; Nieuwenhuis et al., 2005; Raghavanand Cesnik, 2005; Yu et al., 2010). However, no investi-gations have been reported to consider the effects ofthe prestress upon GW mode-tuning capabilities.Figure 7 demonstrates the normalized piezo-actuatedresponses of the fundamental S0 and A0 Lamb wavemodes as a function of sweeping frequencies in theplate under various prestresses. In this figure, e�x x, dð Þ isthe normalized surface strain along the x directiondefined as the ratio of the surface strain with the pres-tress to that without the prestress. The actuator/hostplate geometry parameters are a/h = 20 and d/a =0.333. It can be observed that the overall shape of

wave-tuning curves is retained for both the S0 and A0modes throughout the sweeping frequency under eitherpretension or precompression. The amplitude of the S0mode is almost not affected by the applied prestress asshown in Figure 7(a), while the appearance of the pre-tension or the precompression can lead to an observa-ble change in the wave amplitude and the tuningfrequency shift of the A0 mode as shown in Figure7(b). For example, compared with the case withoutprestress, around 15-kHz variation of tuning frequencyfor the first maximum A0 mode can be observed whenthe applied prestress reaches 0:8sY . Appropriate strate-gies need to be developed to compensate for the influ-ences of the prestress (Croxford et al., 2007), such that

Figure 5. Phase velocities of Lamb wave propagation in an aluminum alloy plate under various prestresses: (a) the S0 mode and (b)the A0 mode.

Song et al. 605

the desired wave mode generation can be achieved tointerrogate the prestressed structure.

Transient GW propagation

In practical GW-based SHM systems, the integratedpiezo-actuators are often excited by narrow-band tone-burst signals to suppress the dispersion of the inducedGWs, which is beneficial for further signal interpreta-tion (Ihn and Chang, 2008). Based on the developedmodel, the GW signals excited by the piezo-actuator inthe prestressed plate are evaluated. Figure 8 shows thetransient piezo-actuated GW responses in the pre-stressed plate under different prestresses. The five-peaktone-burst signal (Song et al., 2012) at a central fre-quency of 150 kHz is used for excitation. The samematerial properties and geometrical parameters inFigure 7 are used for the host medium and the actua-tor. In Figure 8, e�x x, dð Þ is the normalized surface strainalong the x direction defined as the ratio of the surfacestrain with the prestress to that without the prestress.From Figure 8, it can be further substantiated that theS0 mode signal is not sensitive to the prestresses com-pared to the A0 mode, which is good for health moni-toring of prestressed structures utilizing the S0 mode.In contrast, an obvious shift in the wave velocityand amplitude can be found for the A0 mode withvarying prestresses, and such an observed shift hascomparable magnitude to that caused by temperaturevariations (Lee et al., 2010; Raghavan and Cesnik,2008). Similar effects of prestresses on piezo-actuatedGW signals at higher frequency (fc = 400 kHz) can be

seen in Figure 9, where the S0 mode is dominantlyexcited over the A0 mode. It is known that manysignal-processing algorithms (e.g. using damage-induced scattered waves) are based on accurately mea-suring the wave velocity of the interrogative GW sig-nals (Francis Rose and Wang, 2010; Michaels, 2008).Therefore, the influences of the prestress upon GWresponses should be considered to obtain the calibrateddiagnostic image of the structure, especially when theA0 mode is excited for damage detection in the pre-stressed structure.

According to the transient GW responses predicted bythe current model, an appropriate SDC is proposed toquantitatively evaluate the overall change in the receivedGW signals caused by the prestress. The SDC, based onthe root-mean-square change of the GW signals in thetime domain (Michaels et al., 2011), is defined as

SDC s0� �

=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

S tn,s0ð Þ � S tn,s0 = 0ð Þ½ �2,X

n

S tn,s0 = 0ð Þ½ �2vuut

ð23Þ

where S tn,s0ð Þ is the received transient GW signal at

time tn and the applied prestress s0.

As an example, Figure 10 displays the normalizedSDC for both GW modes as a function of appliedprestresses, where SDC* represents the normalizedvalue of SDC with respect to the maximum value. Inthe calculation, the excitation frequency is selected tobe fc = 150 kHz. It can be found, in Figure 10, that the

Figure 6. Effects of structural material properties on prestress-induced velocity variation of the guided wave modes.


shape of the normalized SDC is approximately linearwhen the applied prestress is relatively small (e.g.s0\0:2sY ); however, the variation of the normalizedSDC curve evidently exhibits some nonlinearity for thelarger magnitude of the applied prestress especially forthe A0 mode. Similar variation with respect to appliedloads has been reported in a recent experiment studyon load impact on the GW-based SHM (Michaels etal., 2011). Noticeably, most modern GW methods relyon scattered signals to identify small structural damage,which is often implemented by comparing received sig-nals to baseline signals collected from the undamagedstructure (Huang et al., 2010b; Lin and Yuan, 2005). If

the effects of the prestress on the GW-based SHM areneglected, however, those detection methods may mostlikely provide a false alarm or fail to diagnose in thepresence of applied loads on the monitored structure.

Conclusion

In the study, we focus on the quantitative characteriza-tion of the GW generation and propagation in pre-stressed plate structures induced by the piezoelectricwafer. An analytical model considering coupled piezo-elastodynamics is first developed to study dynamic loadtransfer between a surface-bonded thin piezoelectric

Figure 7. Piezo-actuated responses of the fundamental guided wave modes in a 2-mm thick aluminum alloy plate under variousprestresses: (a) the S0 mode and (b) the A0 mode (actuator/host plate geometry: a/h = 20 and d/a = 0.333).

Song et al. 607

actuator and a prestressed plate. The accuracy of theanalytical prediction is then evaluated through thecomparison with the FE analysis. Based on the currentmodel, the effects of the prestress on the GW propaga-tion and mode-tuning capabilities are investigated atdifferent frequencies for different host materials.Specifically, it is found that (1) the presence of theapplied prestress can result in the observable variationin the wave amplitude and the tuning frequency shift ofthe A0 mode, and very little for the S0 mode; (2) forthe softer host material under the applied prestress, the

variation in the wave velocity becomes more profound;and (3) the normalized SDC curve shows some non-linear variation with respect to the applied prestressespecially for the A0 mode. This study can provide atheoretical base for on-line piezo-GW-based healthmonitoring of the prestressed aerospace structures.

Funding

This research was partly supported by National NaturalScience Foundation of China under Grants No. 10832002and NASA EPSCoR RID grant.

Figure 9. The transient guided wave responses at x = 150 mm away from the piezo-actuator at a central frequency of fc = 400 kHzcollected from the aluminum alloy plate submitted to different prestresses.

Figure 8. The transient guided wave responses at x = 320.0 mm away from the piezo-actuator at a central frequency of fc = 150kHz collected from the aluminum alloy plate submitted to different prestresses.


References

Akbarov SD (2007) Recent investigations on dynamic prob-lems for an elastic body with initial (residual) stresses

(review). International Applied Mechanics 43: 1305–1324.An Y-K and Sohn H (2010) Instantaneous crack detection

under varying temperature and static loading conditions.Structural Control and Health Monitoring 17: 730–741.

Anton SR, Inman DJ and Park G (2009) Reference-free dam-

age detection using instantaneous baseline measurement.American Institute of Aeronautics and Astronautics Journal

47: 1952–1964.Chen F and Wilcox PD (2007) The effect of load on guided

wave propagation. Ultrasonics 47: 111–122.Crawley EF and De Luis J (1987) Use of piezoelectric actua-

tors as elements of intelligent structures. American Instituteof Aeronautics and Astronautics Journal 25: 1373–1385.

Croxford AJ, Wilcox PD, Drinkwater BW, et al. (2007) Stra-

tegies for guided wave structural health monitoring. Pro-ceedings of the Royal Society A: Mathematical, Physical

and Engineering Sciences 463: 2961–2981.Desmet C, Kawald U, Mourad A, et al. (1996) The behavior

of Lamb waves in stressed polymer foils. Journal of theAcoustic Society of America 100: 1509–1513.

Doyle D, Zagrai A, Arritt B, et al. (2010) Damage detectionin bolted space structures. Journal of Intelligent Material

Systems and Structures 21: 251–264.Dugnani R (2009) Dynamic behavior of structure-mounted

disk-shape piezoelectric sensors including the adhesivelayer. Journal of Intelligent Material Systems and Struc-tures 20: 1553–1564.

Francis Rose LR and Wang CH (2010) Mindlin plate theory

for damage detection: imaging of inhomogeneties. Journalof the Acoustic Society of America 127: 754–763.

Fromme P and Sayir MB (2002) Measurement of scattering

of a Lamb wave by a through hole in a plate. Journal of

the Acoustic Society of America 111: 1165–1170.Gangadharan R, Murthy CRL, Gopalakrishnan S, et al. (2009)

Time reversal technique for health monitoring of metallic

structure using Lamb waves. Ultrasonics 49: 696–705.Giurgiutiu V (2005) Tuned Lamb wave excitation and detec-

tion with piezoelectric wafer active sensors for structural

health monitoring. Journal of Intelligent Material Systems

and Structures 16: 291–305.Giurgiutiu V (2008) Structural health monitoring with piezo-

electric wafer active sensors. Amsterdam and Boston, MA:

Elsevier Academic Press.Giurgiutiu V and Santoni GB (2009) Extension of the shear-

lag solution for structurally attached ultrasonic active sen-

sors. American Institute of Aeronautics and Astronautics

Journal 47: 1980–1983.Glushkov E, Glushkova N, Kvasha O, et al. (2007) Integral

equation based modeling of the interaction between piezo-

electric patch actuators and an elastic substrate. Smart

Materials and Structures 16: 650–664.Han L, Wang XD and Zuo M (2009) The dynamic behavior

of a surface bonded piezoelectric actuator with a bonding

layer. Acta Mechanica 206: 193–205.Hu YT, Yang JS and Jiang Q (2002) A model of electroelastic

plates under biasing fields with applications in buckling

analysis. International Journal of Solids and Structures 39:

2629–2642.Huang GL, Song F and Reddy R (2010a) Quantitative mod-

eling of dynamic behavior of a piezoelectric wafer actuator

bonded to an elastic plate for guided wave propagation.

In: Proceedings of SPIE health monitoring of structural and

biological systems, April 2008, San Diego, CA.

Figure 10. SDC* for the guided-wave responses as a function of the applied prestresses.SDC*: normalized signal difference coefficients.

Song et al. 609

Huang GL, Song F and Wang XD (2010b) Quantitative mod-eling of coupled piezo-elastodynamic behavior of piezoelec-

tric actuators bonded to an elastic medium for structuralhealth monitoring: a review. Sensors 10: 3681–3702.

Huang GL and Sun CT (2006) The dynamic behavior of a

piezoelectric actuator bonded to an anisotropic elasticmedium. International Journal of Solids and Structures 43:

1291–1307.Ihn JB and Chang F-K (2008) Pitch-catch active sensing

methods in structural health monitoring for aircraft struc-tures. Structural Health Monitoring 7: 5–19.

Jin C and Wang XD (2011) The effect of adhesive layers on

the dynamic behavior of surface-bonded piezoelectric sen-sors with debonding. Journal of Intelligent Material Sys-

tems and Structures 22: 655–670.Karami G, Grundman N, Abolfathi N, et al. (2008) Thermal

elastic characterization and evaluation of residual stresses

in bi-directional fibrous composites. Applied CompositeMaterials 15: 259–272.

Kundu T and Maslov K (1997) Material interface inspectionby Lamb waves. International Journal of Solids and Struc-

tures 34: 3885–3901.Kwun H, Bartels KA and Hanley JJ (1998) Effects of tensile

loading on the properties of elastic-wave propagation in a

strand. Journal of the Acoustic Society of America 103:3370–3375.

Lanza di Scalea F and Salamone S (2008) Temperature

effects in ultrasonic Lamb wave structural health monitor-ing systems. Journal of the Acoustic Society of America124: 161–174.

Lee SJ, Gandhi N, Michaels JE, et al. (2010) Comparison of

the effects of applied loads and temperature variations onguided wave propagation. AIP Conference Proceedings

1335: 175–182.Lesieutre GA (2009) How membrane loads influence

the modal damping of flexural structures. AmericanInstitute of Aeronautics and Astronautics Journal 47:

1642–1646.Lin X and Yuan F-G (2005) Experimental study applying a

migration technique in structural health monitoring. Struc-

tural Health Monitoring 4: 341–353.Liu H, Kuang ZB and Cai ZM (2003) Propagation of Bleus-

tein–Gulyaev waves in a prestressed layered piezoelectric

structure. Ultrasonics 41: 397–405.Loveday PW (2009) Semi-analytical finite element analysis of

elastic waveguides subjected to axial loads. Ultrasonics 49:298–300.

Michaels JE (2008) Detection, localization and characteriza-

tion of damage in plates with an in situ array of spatiallydistributed ultrasonic sensors. Smart Materials and Struc-

tures 17: 035035.Michaels JE, Lee SJ and Michaels TE (2011) Impact of

applied loads on guided wave structural health monitor-

ing. AIP Conference Proceedings 1335: 1515–1522.Nieuwenhuis JH, Neumann JJ Jr, Greve DW, et al. (2005)

Generation and detection of guided waves using PZTwafer transducers. IEEE Transactions on Ultrasonics, Fer-

roelectrics 52: 2103–2111.Park S, Anton SR, Kim JK, et al. (2010) Instantaneous base-

line structural damage detection using a miniaturized

piezoelectric guided waves system. KSCE Journal of Civil

Engineering 14: 889–895.Raghavan A and Cesnik CES (2005) Finite-dimensional

piezoelectric transducer modeling for guided wave based

structural health monitoring. Smart Materials and Struc-

tures 14: 1448–1461.Raghavan A and Cesnik CES (2007) Review of guided wave-

structural health monitoring. Shock and Vibration Digest

39: 91–114.Raghavan A and Cesnik CES (2008) Effects of evaluated tempera-

ture on guided wave structural health monitoring. Journal of

Intelligent Material Systems and Structures 19: 1383–1398.Ratwani MM (2000) Repair options for airframes. In: Aging

aircraft fleets—structural and other subsystem aspects,

Sofia, Bulgaria, AGARD Lecture Series 218.Santus C and Tayler D (2009) Physically short crack propaga-

tion in metals during high cycle fatigue. International Jour-

nal of Fatigue 31: 1356–1365.Senesi M and Ruzzene M (2011) A frequency selective acous-

tic transducer for directional Lamb wave sensing. Journal

of the Acoustic Society of America 130: 1899–1907.

Singh B (2010) Wave propagation in a prestressed piezoelec-

tric half space. Acta Mechanica 211: 337–344.Song F, Huang GL and Hudson K (2009) Guided wave pro-

pagation in honeycomb sandwich structures by using a

piezoelectric actuator/sensor system. Smart Materials and

Structures 18: 125007.Song F, Huang GL and Hu GK (2012) On-line guided wave

based debonding detection in honeycomb sandwich struc-

tures. American Institute of Aeronautics and Astronautics

Journal 50: 284–293.Su J, Kuang ZB and Liu H (2005) Love wave in ZnO/SiO2/Si

structure with initial stresses. Journal of Sound and Vibra-

tion 286: 981–999.Tiersten HF, Sinha BK and Meeker TR (1981) Intrinsic stress

in thin films deposited on anisotropic substances and its

influence on the natural frequencies of piezoelectric reso-

nators. Journal of Applied Physics 52: 5614–5624.Wang J, Wu RX, Du JK, et al. (2007) Effect of initial stresses

on surface acoustic waves propagating in infinite elastic

plates. In: IEEE international frequency control symposium,

May, Geneva, Switzerland.Wang XD and Huang GL (2001) Wave propagation in elec-

tromechanical structures: induced by surface-bonded

piezoelectric actuators. Journal of Intelligent Material Sys-

tems and Structures 12: 105–115.Yang J (2005) Free vibrations of an electroelastic body under

biasing fields. IEEE Transactions on Ultrasonics, Ferroelec-

trics 52: 358–364.Yu Y, Santoni GB and Giurgiutiu V (2010) Shear lag solu-

tion for tuning ultrasonic piezoelectric wafer active sensors

with applications to Lamb wave array imaging. Interna-

tional Journal of Engineering Science 48: 848–861.Von Ende S and Lammering R (2007) Investigation on piezo-

electrically induced Lamb wave generation and propaga-

tion. Smart Materials and Structures 16: 1802–1809.Zagrai A, Doyle D, Gigineishvili V, et al. (2010) Piezoelectric

wafer active sensor structural health monitoring of space

structures. Journal of Intelligent Material Systems and

Structures 21: 921–940.


Appendix 1

Determination of the resulting initial stress

When a static stress s0 is initially applied on the hostplate, the actuator is submitted to the resulting axialinitial stress ŝ0. Based on the assumptions in the sub-section ‘Modeling of the prestressed piezoelectric actua-tor,’ the actuator can be modeled as an electroelasticline subjected to the distributed axial load t0

�h, and

the equilibrium equation of the actuator along the axialdirection can be then written as

dŝ0

dx+

t0 xð Þh

= 0: ð24Þ

Since the load transferred between the actuator andthe host plate can be attributed to t0, the two ends ofthe actuator can be assumed to be traction free, that is

ŝ0 = 0, xj j= a ð25Þ

By integrating equation (24) and making use ofequation (25), the axial stress in the actuator can beexpressed in terms of the shear stress t0 as

ŝ0 xð Þ= �ðx�a

t0 zð Þh

dz, xj j= a ð26Þ

with

ða�a

t0 zð Þ dz = 0 ð27Þ

The relation between the axial initial stress (ŝ0),strain (ê0), and the electric fields (Ê0) of this actuatormodel can be obtained by using the following generalconstitutive relation

ŝ0 =Eaê0 � eaÊ0 ð28Þ

where Ea and ea are the effective elastic and piezoelec-tric material constants, Ea = c11 � c213

�c33 and

ea = e13 � e33c13=c33, respectively, with cij and eij (i, j= 1, 3) being the components of the piezoelectric elas-tic stiffness matrix for a constant electric potential, andthe piezoelectric constant matrix, respectively. In theinitial static deformation, Ê0 = 0 is assumed. Theresulting initial axial strain can then be obtained interms of t0 as

ê0 xð Þ= �ðx�a

t0 zð ÞhEa

dz, xj j\a ð29Þ

The stress field produced inside the host plate can becaused by both the initial stress applied on the hostplate with a traction free boundary and the surfaceshear stress t0 resulted from the actuator. The straininduced by t0 can be obtained by using the boundarycondition along the top surface as

s0xz x, dð Þ= � t0 xð Þ, xj j\a and s0zz x, dð Þ= 0 ð30Þ

Making use of the quasi-static fundamental solutionof an elastic plate subject to a tangential traction andfollowing a similar manner as described in Huang et al.(2010a), the strain ~e0x resulting from the applied traction(equation (30)) can be determined in terms of the staticinterfacial shear stress t0. Together with the constantstrain e0 =s0

�E applied along the x direction at infi-

nity, the total surface strain in the host plate can thenbe found by superimposing the solutions of both partsas e0x x, dð Þ= ~e0x + e0.

The continuity of deformation between the actuatorand the host medium indicates that

e0x x, dð Þ= ê0, xj j\a ð31Þ

Expanding t0 in terms of Chebyshev polynomialsand following a similar procedure with the ‘Dynamicload transfer and resulting GW propagation’ subsec-tion, the static interfacial shear stress t0 can be solved.Therefore, based on equation (26), the axial stress ŝ0 xð Þin the actuator can be obtained.

Song et al. 611

Documents

DOI: 10.1177/1045389X12467516 and propagation in plates with … · 2019. 5. 29. · modeling of guided wave excitation and propagation in plates with applied prestresses F. Song1,