Accepted Manuscript Analysis of second-harmonic generation by primary ultrasonic guided wave propagation in a piezoelectric plate Mingxi Deng, Yanxun Xiang PII: DOI: Reference:
S0041-624X(15)00097-9 http://dx.doi.org/10.1016/j.ultras.2015.04.005 ULTRAS 5033
To appear in:
Ultrasonics
Received Date: Revised Date: Accepted Date:
2 September 2014 2 March 2015 9 April 2015
Please cite this article as: M. Deng, Y. Xiang, Analysis of second-harmonic generation by primary ultrasonic guided wave propagation in a piezoelectric plate, Ultrasonics (2015), doi: http://dx.doi.org/10.1016/j.ultras.2015.04.005
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Analysis of second-harmonic generation by primary ultrasonic guided wave propagation in a piezoelectric plate
Mingxi Deng a,*, Yanxun Xiang b a)
a
Department of Physics, Logistics Engineering University, Chongqing 401331, China
b
School of Mechanical and Power Engineering, East China University of Science and
Technology, Shanghai 200237, China
ABSTRACT The effect of second-harmonic generation (SHG) by primary ultrasonic guided wave propagation is analyzed, where the nonlinear elastic, piezoelectric, and dielectric properties of the piezoelectric plate material are considered simultaneously. The formal solution of the corresponding second-harmonic displacement field is presented. Theoretical and numerical investigations clearly show that the SHG effect of primary guided wave propagation is highly sensitive to the electrical boundary conditions of the piezoelectric plate. The results obtained may provide a means through which the SHG efficiency of ultrasonic guided wave propagation can effectively be regulated by changing the electrical boundary conditions of the piezoelectric plate.
Keywords: Ultrasonic guided waves; Piezoelectric plate; Second-harmonic generation (SHG); Electrical boundary condition
*
Corresponding author. Tel.: +86 23 86730953. E-mail addresses: [email protected] (M.X. Deng), [email protected] (Y.X. Xiang). 1
2nd Revised manuscript submitted to Ultrasonics
1. Introduction Ultrasonic guided waves propagationg in layered piezoelectric plates have been applied in practical applications such as acoustic sensing, ultrasonic nondestructive evaluation, and signal processing, etc [1-3]. Generally, there are two sources of nonlinearity in primary guided wave propagation, namely, the convective nonlinearity independent of the material properties, and that produced by the nonlinear elastic, piezoelectric, and dielectric properties of the piezoelectric materials [4-6]. So, nonlinear effects will occur in conjunction with primary ultrasonic guided wave propagation in piezoelectric plates. One of the typical nonlinear effects is the generation of second harmonics. In recent years the studies on the effect of second-harmonic generation (SHG) by primary ultrasonic guided wave propagation attract more and more attention because of its potential to accurately assess the mechanical properties of plate-like structures [7-12]. However, the present studies on the SHG effect of ultrasonic guided waves are conducted only for the case where the materials of plate-like structures are assumed to be non-piezoelectric, and only the elastic nonlinearity of materials is taken into account [13-17]. It can be expected that besides the elastic nonlinearity in pizeoelectric materials, the piezoelectric and dielectric nonlinearities will likewise contribute to the SHG effect of primary guided wave propagation [5,6]. Moreover, besides the dispersion and multi-mode characteristics in guided wave propagation [13-17], the inherent coupling between the electric and mechanical fields in pizeoelectric materials will further increase the complexity of analyzing the SHG effect. Considering the potentials of using the SHG effect in practical applications [7-12], it is essential to analyze the SHG effect of primary ultrasonic 2
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guided wave propagation in piezoelectric plates, where the nonlinearities in elasticity, piezoelectricity, and dielectricity will be considered simultaneously. The aim of the present paper is to establish a model that can be used to analyze the said SHG effect. Especially, the analyses will focus on the influences of the electrical boundary conditions on the SHG effect of ultrasonic guided wave propagation in piezoelectric plates.
2. Theoretical fundamentals For simplicity, we just analyze ultrasonic guided wave propagation in a single piezoelectric plate. The piezoelectric material is assumed to be homogeneous with no attenuation and no dispersion. However, the present model can readily be extended for application to more complex piezoelectric plate-like structures. The Lagrangian coordinate system a1a2a3 established for a single piezoelectric plate is shown in Fig. 1, where γ1, γ2 and
γ3 are the angles of orientation between the two coordinate systems a1a2a3 and a1′a2′ a3′ , and the elastic, piezoelectric and dielectric constants of the piezoelectric material are originally defined under the coordinate system a1′a2′ a3′ . For a sound wave propagating along the Oa3 axis in Fig. 1, its formal solution is given by ui exp[j( ka3 + α ka2 − ω t )] under quasi-static approximation [18], where ui (i=1,2,3) corresponds to the mechanical displacement component along the Oai axis, and ui (i=4) the electrical potential. Based on the Christoffel equation for plane wave propagation in piezoelectric solids [18], the eight roots of α (denoted by α(q), q=1,…,8) can be obtained, and further, the corresponding eigensolution (u1( q ) , u2( q ) , u3( q ) ,ϕ ( q ) ) can also be determined for each
α(q). For notational simplicity, the Einstein summation convention is used throughout. 3
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Generally, a linear combination of eight eigensolutions (u1( q ) , u2( q ) , u3( q ) ,ϕ ( q ) ) constitutes the general solution of wave motion (neglecting the factor exp[-jωt]) [18] U i = Aqui( q ) exp[j( ka3 + α ( q ) ka2 )] , ϕ = Aqϕ ( q ) exp[j( ka3 + α ( q ) ka2 )],
(1)
where Ui (i=1,2,3) is the particle displacement component along the Oa1, Oa2 or Oa3 axis, and
ϕ the electrical potential, and Aq (q=1,…,8) are arbitrary constants to be determined. When a guided wave propagation is postulated, the corresponding dispersion equation can be obtained from the mechanical (stress-free at a2=0, h) and electrical (short-circuit or open circuit at a2=0, h) boundary conditions [18]. After that, the ratio of Aq to A1 can be determined by substituting the frequency-thickness product fh and the corresponding guided wave phase velocity into the equations of the mechanical and electrical boundary conditions [15]. The formal solution of a guided wave mode propagating in the piezoelectric plate (see Fig.1) has the same form as that shown in Eq. (1). Due to the anisotropy of the piezoelectric material, the guided wave propagation cannot generally be decoupled into two independent wave modes: Lamb wave and shear horizontal plate wave [15,18]. It should be noted here that the short-circuit boundary condition at the surface of piezoelectric plate (a2=0 or h in Fig. 1) means that the corresponding electrical potential at a2=0 or h is zero, while the open circuit boundary condition at the surface of piezoelectric plate (a2=0 or h) means that the corresponding surface charge density at a2=0 or h is zero [18]. Generally, the analysis of the SHG effect of primary guided wave propagation is extremely complicated due to its dispersion and multi-mode characteristics [13-17]. However, the elastic, piezoelectric and dielectric nonlinearities of piezoelectric materials are often small and the successive approximation approach can be applied [4-6]. Second-order perturbation 4
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reduces a complicated nonlinear problem to the linear one with known excitation sources, so that the linear analysis approach can be applied even if the original problem is nonlinear [13,14]. When the primary (f, p) guided wave (with the driving frequency f and the order p) propagates along the direction of the Oa3 axis in Fig. 1, in the interior of the solid plate, there are bulk driving force components with the frequency 2f, denoted by [4-6] fi ( 2 f ) = M ijklmn
∂ 2U k ∂U m ∂ 2ϕ ∂ϕ − d klij ∂a j ∂al ∂an ∂a j ∂ak ∂al
∂ 2U l ∂ϕ ∂U l ∂ 2ϕ + ( f kijlm + δ ilekmj ) + ∂a j ∂am ∂ak ∂am ∂a j ∂ak
,
(2)
where Mijklmn is defined by cijklmn+ δkmcijln+ δimcjnkl+δikcjlmn (i, j, k, l, m, n=1,2,3), and cijkl and cijklmn are second- and third-order elastic constants. The constant ekmj is second-order piezoelectric one. It should be noted that the electrostrictive constant dklij is related both to elasto-optic and electrostrictive effects, and that the third-order piezoelectric constant fkijlm is related to the electroacoustic effect [5,6]. Unlike wave motion in non-piezoelectric material, there is a bulk charge density with the frequency 2f, denoted by [5,6]
ρ ( 2 f ) = ε ijk
∂ 2ϕ ∂ϕ ∂ 2U j ∂U l + ( fijklm + δ jl eikm ) ∂ai∂a j ∂ak ∂ai ∂ak ∂am
∂ 2ϕ ∂U k ∂ϕ ∂ 2U k − d ijkl + ∂ai∂a j ∂al ∂a j ∂ai ∂al
,
(3)
where εijk is third-order dielectric constant. It is noted that all the material constants in Eqs. (2) and (3) should be transformed from the coordinate system a1′a2′ a3′ to a1a2 a3 [15,18,19]. Besides f i ( 2 f ) and ρ(2f) in the interior of the piezoelectric plate, there are second-order stress tensors (with the frequency 2f ) at the two surfaces of the piezoelectric plate, denoted by [4-6]
Pij( 2 f ) =
∂U k ∂U m 1 ∂ϕ ∂ϕ ∂U ∂ϕ 1 M ijklmn − d klij + ( f kijlm + δ il ekmj ) l , 2 ∂al ∂an 2 ∂ak ∂al ∂am ∂ak
and second-order surface charge density, denoted by [5,6] 5
(4)
2nd Revised manuscript submitted to Ultrasonics
1 2
σ ( 2 f ) = ε 2 jk
∂ϕ ∂ϕ 1 ∂U ∂U l ∂ϕ ∂U k . + ( f 2 jklm + δ jl e2 km ) j − d 2 jkl ∂a j ∂ak 2 ∂ak ∂am ∂a j ∂al
(5)
The second-order terms, f i ( 2 f ) , ρ(2f), Pij( 2 f ) , and σ(2f), can provide a complete and accurate description for the SHG effect [5,6]. According to the modal analysis approach for waveguide excitation [13,14,18], these second-order terms can be regarded as excitation sources for the generation of a series of double frequency guided wave (DFGW) modes. Thus, the field of the SHG by the (f, p) guided wave propagation can formally be expressed by the summation of these DFGWs, i.e. [18], U i( 2) = α s (a3 ) ⋅ U i( 2 f ,s ) (a2 ) , ϕ ( 2) = α s (a3 ) ⋅ ϕ ( 2 f ,s ) (a2 ) ,
(6)
where U i( 2 f ,s ) (a2 ) (i=1,2,3) and ϕ(2f,s)(a2) are, respectively, the Oai-axis displacement component and the electrical potential of the DFGW mode (with the driving frequency 2f and the order s), and α s (a3 ) is the corresponding expansion coefficient. Based on the reciprocity relation and the orthogonality of guided waves, the equation governing αs(a3) is given by [13-18], FS + Fb ∂ ( 2 f ,s ) ∂a − j k α s (a3 ) = 4 P , 3 ss
(7)
where Pss is the average power flow per unit width along the Oa1 axis for the sth DFGW mode, and k(2f,s) is the Oa3-axis component of the wave vectors. The forcing function due to the bulk sources f i ( 2 f ) and ρ(2f) is given by [5,6,18] h ~ h Fb= 2 jω ∫0 [U i( 2 f ,s ) (a2 ) ⋅ f i ( 2 ) ] d a2 − j 2ω ∫0 [ϕ~ ( 2 f ,s ) (a2 ) ⋅ ρ ( 2) ] d a2 ,
(8)
and, after considerations of the mechanical and electrical boundary conditions at the two surfaces a2=0 and h, the forcing function due to the surface sources Pij( 2 f ) and σ(2f) is derived as follows [5,6,18] 6
2nd Revised manuscript submitted to Ultrasonics a2 = h ~ a2 = h FS = 2 jω[U i( 2 f , s ) (a2 ) ⋅ Pi 2( 2 f ) ] + 2 jω[ϕ~ ( 2 f ,s ) (a2 ) ⋅ σ ( 2 f ) ] a2 =0 ,
(9)
a2 = 0
where the sign ‘~’ on the top of U i( 2 f ,s ) and ϕ ( 2 f ,s ) denotes the complex conjugate operation. For the short-circuit or open circuit case at a2=0 or h, the second term in the right-hand side of Eq. (9) should, respectively, be removed or reserved. Considering an initial condition for a DFGW generation, i.e., αs(a3)=0 when a3=0, αs(a 3) can formally be given by [13,14]
α s (a3 ) =
α s sin[ D ka3 ] 4
D kh
exp[j k ( 2 f ,s ) a3 + j D ka3 ] ,
(10)
where αs=(FS+Fb)h exp(-2jka 3)/Pss; c(f,p) = ω/k and c(2f,s)=2ω/k(2f,s) are, respectively, the phase velocity of the (f, p) guided wave and the sth DFGW mode; the parameter D=[c(2f,s)-c(f,p)]/c(2f,s) can be used to describe the degree of dispersion (i.e., the relative difference of phase velocity). The solution given in Eq. (10) is exact within the second-order perturbation approximation. When αs ≠ 0, and the phase velocity matching condition between the primary mode and the DFGW [i.e., D=0 in Eq. (10)] is satisfied, αs(a3) can further be written as
αs(a3)=
α s a3 4
⋅
h
exp[j k ( 2 f ,s ) a3 ] ,
(11)
meaning that the sth DFGW component will increase with propagation distance. When αs ≠ 0 and D ≠ 0, there will be a beat effect for the amplitude of the sth DFGW with propagation distance, and the contribution of sth DFGW to U i( 2 f ) can be negligible [13]. These results are similar to that when a guided wave propagates in a non-piezoelectric plate [13-15]. Provided that the piezoelectric and dielectric nonlinearities of material are not considered (namely, the constants ekmj, dklij, and fkijlm equal zero), ρ(2) and σ(2f) in Eqs. (3) and (5) will vanish, and the mathematical expression of f i( 2 f ) and Pij( 2 f ) in Eqs. (2) and (4), as well as
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the solution of the DFGWs shown in Eqs. (8)-(10), will reduce to the same results as that found earlier in Refs. [13-15]. Thus, the analysis model proposed in this paper is more general than that derived previously. For a given set of orientation angles (γ1, γ2, γ3), the primary (f, p) guided wave propagating in the piezoelectric plate with the specific electrical boundary conditions at a2=0 and h (see Fig. 1) can be selectively generated to ensure αs ≠ 0 and D=0. In this way, the sth DFGW component will grow with propagation distance, and the obvious second-harmonic signals of primary guided wave propagation can be observed. When the changes in the electrical boundary conditions take place, the SHG effect of primary ultrasonic guided wave propagation will be influenced in the following two aspects. First, changes in the electrical boundary conditions will influence the dispersion relations of guided waves [18]. Thus, the original phase velocity matching condition D=0 [i.e., c(f,p)= c(2f,s)] may not be satisfied now when the electrical boundary conditions change. This will remarkably influence the efficiency of the SHG by the (f, p) guided wave propagation [13]. Second, changes in the electrical boundary conditions may provide different acoustic field features for the (f, p) guided wave. This will influence the magnitude of αs(a 3) because FS and Fb in Eq. (10) are proportional to the square of amplitude of the (f, p) guided wave [4-6].
3. Numerical analyses In the previous section, a theoretical model has been established for the SHG by primary ultrasonic guided wave propagation in a piezoelectric plate. Based on the model established, the possible influences of the electrical boundary conditions on the SHG have been 8
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qualitatively analyzed. Next some numerical analyses are given to understand the above theoretical analyses. The plate material is assumed to be single crystal LiNbO3 (Trigonal system, 3m), and its parameters are given in Refs. [19] and [20]. Under the open circuit conditions at a2=0 and h, the dispersion curves for the primary guided wave mode and the DFGWs propagating in the LiNbO3 plate with (γ1, γ2, γ3) = (0 o, 0o, 0o) are computed and shown in Fig. 2. The intersections between the vertical dotted line V (fh=4.75MHz.mm) and the dispersion curves of the DFGWs (denoted by points s0, s1, s2, s3 and s4, etc.) denote the DFGW mode components constituting the field of the second harmonic of the primary guided wave (denoted by point p 0) propagation. At this given frequency-thickness product fh=4.75MHz.mm, points p0 and s0 overlap, meaning that the primary guided wave (point p 0) and the DFGW (points s0) have the same phase velocity, i.e., c ( f , p0 ) = c ( 2 f , s0 ) = 7.364MHz.mm. Some parameters of the DFGWs (determined by points s0, s1, s2, s3 and s4, etc.) generated by the primary guided wave (point p0) propagation are listed in Table I. Fig. 3 shows the curves of amplitudes of these DFGW components (at the surface a2=h) versus propagation distance. Clearly, the amplitude of the DFGW (point s0) at the surface a 2=h grows linearly with propagation distance due to D=0 and
αs ≠ 0 in Eq. (10). For the other DFGW components determined by points s1, s2, s3 and s4, etc., their amplitudes exhibit a beat effect (oscillate) with propagation distance because of the phase velocity mismatching between the primary mode and the DFGW [i.e., D ≠ 0 in Eq. (10)] [13]. From the viewpoint of practical measurement, the DFGW component (point s0) plays a dominant role in the second-harmonic field of the primary guided wave mode (point p0), but 9
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the contribution of the other DFGW components (determined by points s1, s2, s3 and s4, etc.) to both U i( 2) and ϕ ( 2) in Eq. (6) can be neglected after the primary guided wave has propagated some distance [13]. So it can be regarded that the field of the SHG is only determined by the DFGW mode (point s0). Next, the analyses will focus on the influences of the electrical boundary conditions on the generation of the sth DFGW. Generally speaking, changes in the electrical boundary conditions will lead to changes in the phase velocity of guided waves, as well as FS, Fb, and
αs in Eqs. (7) and (10). When the orientation angles (γ1, γ2, γ3)= (0o, 0o, 0o) are kept unchanged, the phase velocity curves versus fh under different electrical boundary conditions are shown in Fig. 4. When the electrical boundary conditions change from the 1 st case (open circuit at a2=0 and h) to the 2nd case (open circuit at a2=0 and short-circuit at a2= h ) or the 3 rd case (short-circuit at a2=0 and h ), the mode pair of the primary and DFGW modes (denoted by intersections between the vertical dotted line V and the dispersion curves in Fig. 4) changes from point pair (p0, s0) to ( p ' , s ' ) or ( p ' ' , s ' ' ). Clearly, points p ' and s ' , as well as p ' ' and s' ' in Fig. 4, no longer overlap, meaning that there is a phase velocity mismatching between
the primary mode and the DFGW (i.e., D ≠ 0). When the frequency-thickness product fh is given by the vertical dotted line V (fh =4.75MHz.mm), some parameters of the sth DFGW (denoted, respectively, by point s0, s ' or s ' ' ) generated by the primary guided wave propagation (denoted, respectively, by point p0, p ' or p ' ' ) are listed in Table II. When the electrical boundary conditions change (relative to the open circuit conditions at a 2=0 and h), the amplitudes of the sth DFGW (at the surface a 2=h ) versus propagation distance are shown in Fig. 5. Obviously, the numerical results reveal that, using different 10
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electrical boundary conditions, the SHG effect of the primary guided wave propagation can be effectively regulated, relative to a specific electrical boundary condition that can ensure αs ≠ 0 and D=0 in Eq. (10). Specifically speaking, at the propagation distance given by the vertical dotted line H in Fig. 5, the SHG efficiency (characterized by the sth DFGW amplitude that is determined by the y-axis value of point D0, D1, or D2 ) can be effectively modified by changing the electrical boundary conditions.
4. Conclusions The nonlinearity in ultrasonic guided wave propagation is treated as a second-order perturbation to the linear wave motion response. Based on the modal expansion analysis for waveguide excitation, an accurate description for the SHG effect of primary ultrasonic guided wave propagation in a piezoelectric plate has been presented within a second-order perturbation approximation. The formal solution of the DFGWs, constituting the field of second harmonic, has been developed. The analytical results clearly reveal that the SHG effect of primary guided wave propagation is closely related to the electric boundary conditions of the piezoelectric plate. It is found that under different electrical boundary conditions there is an evident difference in the SHG effect of ultrasonic guided waves, and that the SHG effect is highly sensitive to the electrical boundary conditions. The results obtained may provide a means for regulating the SHG efficiency of ultrasonic guided waves by changing the electrical boundary conditions of piezoelectric plates.
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Acknowledgement The authors are grateful for the support provided by the National Natural Science Foundations of China under Grant Nos. 11274388, 11474093 and 11474361.
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References [1] M.H. Badi, G.G. Yaralioglu, A.S. Ergun, S.T. Hansen, E.J. Wong, B.T. Khuri-Yakub, Capacitive micromachined ultrasonic Lamb wave transducers using rectangular membranes, IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 50 (2003) 1191-1203. [2] J.L. Rose, A baseline and vision of ultrasonic guided wave inspection potential, Journal of Press Vessel Technology, 124 (2002) 273-282. [3] A.H. Nayfeh, Wave Propagation in Layered Anisotropic Media, Elsevier, Amsterdam, 1995. [4] A.N. Norris, in Nonlinear Acoustics, edited by M.F. Hamilton and D.T. Blackstock, Academic, New York, 1998. [5] D.H. McMahon, Acoustic second-harmonic generation in piezoelectric crystals, J. Acoust. Soc. Am. 44 (1968) 1007-1013. [6] V.E. Ljamov, Nonlinear acoustical parameters of pizeoelectric crystals, J. Acoust. Soc. Am. 52 (1972) 199-202. [7] M.X. Deng, Characterization of surface properties of a solid plate using nonlinear Lamb wave approach, Ultrasonics, 44 (2006) e1157-e1162. [8] M.X Deng, J.F. Pei, Assessment of accumulated fatigue damage in solid plates using nonlinear Lamb wave approach, Appl. Phys. Lett. 90 (2007) 121902. [9] C. Bermes, J.Y. Kim, J.M. Qu, L.J. Jacobs, Experimental characterization of material nonlinearity using Lamb waves, Appl. Phys. Lett. 90 (2007) 021901. [10] C. Pruell, J.Y. Kim, J.M. Qu and L.J. Jacobs, Evaluation of fatigue damage using nonlinear guided waves, Smart Mater. Struct. 18 (2009) 035003.
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[11] Y.X. Xiang, M.X. Deng, F. Xuan, Cumulative second-harmonic analysis of ultrasonic Lamb waves for ageing behavior study of modified-HP austenite steel, Ultrasonics, 51(2011) 974-981. [12]. W.B. Li, Y. Cho, J.D. Achenbach, Detection of thermal fatigue in composites by second harmonic Lamb waves, Smart Mater. Struct. 21 (2012) 085019. [13] M.X. Deng, Analysis of second-harmonic generation of Lamb modes using a modal analysis approach, J. Appl. Phys. 94 (2003) 4152-4159. [14] W.J.N. de Lima, M.F. Hamilton, Finite-amplitude waves in isotropic elastic plates, J. Sound. Vib. 265 (2003) 819–839. [15] M.X. Deng, Second-harmonic generation of ultrasonic guided wave propagation in an anisotropic solid plate, Appl. Phys. Lett. 92 (2008) 111910. [16] V.K. Chillara, C.J. Lissenden, Interaction of guided wave modes in isotropic weakly nonlinear elastic plates: higher harmonic generation, Journal of Applied Physics 111 (2012) 124909. [17] V.K. Chillara, C.J. Lissenden, Analysis of second harmonic guided waves in pipes using a large-radius asymptotic approximation for axis-symmetric longitudinal modes, Ultrasonics, 53 (2013) 862-869. [18] B.A. Auld, Acoustic Fields and Waves in Solids, Vols. I and II, Wiley, New York, 1973. [19] D.F. Nelson, Electric, Optic, and Acoustic Interactions in Dielectrics, Wiley, New York, 1979. [20] Y. Cho, K. Yamanouchi, Nonlinear, elastic, piezoelectric, electrostrictive, and dielectric constants of lithium niobate, J. Appl. Phys. 61 (1987) 875-887.
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Table I. Some parameters associated with the SHG at fh=4.75MHz.mm under open circuit conditions.
a
Primary mode
( f , p0 )
( f , p0 )
( f , p0 )
( f , p0 )
( f , p0 )
DFGW
( 2 f , s0 )
(2 f , s1 )
( 2 f , s2 )
( 2 f , s3 )
( 2 f , s4 )
c(f, p ) (MHz.mm)
7.364
7.364
7.364
7.364
7.364
c(2f, s) (MHz.mm)
7.364
7.086
5.078
4.024
9.479
αs a
(16.92, 0.09)
(-0.02, -2.10) (2.32, 0.83) (-1.07, 12.15) (-14.02. 0.65)
The unit or scale of α sU i( 2 f ,s ) (a2 ) (i=1,2,3) is Am2 h , where Am is the amplitude of the
Oa 3-axis displacement component of the (f, p) guided wave propagation at the surface a 2=0.
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Table II. Some parameters associated with the SHG at fh=4.75MHz.mm under different electrical boundary conditions.
a
Primary mode
( f , p0 )
( f , p' )
( f , p' ' )
DFGW
( 2 f , s0 )
(2 f , s ' )
(2 f , s ' ' )
c(f, p) (MHz.mm)
7.364
7.311
7.243
c(2f, s) (MHz.mm)
7.364
7.338
7.296
αs a
(16.92, 0.09)
(23.13, -0.96)
(38.53, -4.00)
The unit or scale of α sU i( 2 f ,s ) (a2 ) (i=1,2,3) is Am2 h , where Am is the amplitude of the
Oa 3-axis displacement component of the (f, p) guided wave propagation at the surface a 2=0.
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Figure Captions
Fig. 1. (Color online) The piezoelectric plate and the Lagrangian coordinate system.
Fig. 2. (Color online) Phase velocity dispersion curves for the primary guided wave and the DFGWs under the open circuit conditions.
Fig. 3. (Color online) Amplitudes of the sth DFGW vs propagation distance under the open circuit conditions: (a) the Oa 2-axis displacement component, and (b) the Oa3-axis displacement component. The Oa 1-axis displacement component vanishes for the case of (γ1,
γ2, γ3) = (0o, 0o, 0o).18
Fig. 4. (Color online) Phase velocity dispersion curves for the primary guided waves and the DFGWs under different electrical boundary conditions.
Fig. 5. (Color online) Amplitudes of the sth DFGW vs propagation distance for different electrical boundary conditions: (a) the Oa2-axis displacement component, and (b) the Oa 3-axis displacement component. The Oa 1-axis displacement component vanishes for the case of (γ1,
γ2, γ3) = (0o, 0o, 0o).18
17
Figure 1
Figure 2
Figure 3(a)
Figure 3(b)
Figure 4
Figure 5(a)
Figure 5(b)
2nd Revised manuscript submitted to Ultrasonics
This paper establishes a model that can be used to analyze the second-harmonic generation (SHG) of primary ultrasonic guided wave propagation in piezoelectric plates. Previous studies are conducted only for the case where the materials of plate-like structures are assumed to be non-piezoelectric, and only the elastic nonlinearity of materials is taken into account. The present work not only takes into account of the effect of elastic nonlinearity of materials, but also considers the nonlinearities in elasticity, piezoelectricity, and dielectricity simultaneously. It has been found out that the electrical boundary conditions of piezoelectric plates have a large influence on the SHG effect of ultrasonic guided wave propagation. 1. A model has been established for analyzing the SHG effect of ultrasonic guided waves in piezoelectric plates. 2. The possible influences of electrical boundary conditions of piezoelectric plates on the SHG effect have been analyzed. 3. The results obtained provide a means through which the SHG efficiency of ultrasonic guided wave propagation can be effectively regulated by changing the electrical boundary conditions of piezoelectric plates.
18
S0041-624X(15)00097-9 http://dx.doi.org/10.1016/j.ultras.2015.04.005 ULTRAS 5033
To appear in:
Ultrasonics
Received Date: Revised Date: Accepted Date:
2 September 2014 2 March 2015 9 April 2015
Please cite this article as: M. Deng, Y. Xiang, Analysis of second-harmonic generation by primary ultrasonic guided wave propagation in a piezoelectric plate, Ultrasonics (2015), doi: http://dx.doi.org/10.1016/j.ultras.2015.04.005
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
2nd Revised manuscript submitted to Ultrasonics
Analysis of second-harmonic generation by primary ultrasonic guided wave propagation in a piezoelectric plate
Mingxi Deng a,*, Yanxun Xiang b a)
a
Department of Physics, Logistics Engineering University, Chongqing 401331, China
b
School of Mechanical and Power Engineering, East China University of Science and
Technology, Shanghai 200237, China
ABSTRACT The effect of second-harmonic generation (SHG) by primary ultrasonic guided wave propagation is analyzed, where the nonlinear elastic, piezoelectric, and dielectric properties of the piezoelectric plate material are considered simultaneously. The formal solution of the corresponding second-harmonic displacement field is presented. Theoretical and numerical investigations clearly show that the SHG effect of primary guided wave propagation is highly sensitive to the electrical boundary conditions of the piezoelectric plate. The results obtained may provide a means through which the SHG efficiency of ultrasonic guided wave propagation can effectively be regulated by changing the electrical boundary conditions of the piezoelectric plate.
Keywords: Ultrasonic guided waves; Piezoelectric plate; Second-harmonic generation (SHG); Electrical boundary condition
*
Corresponding author. Tel.: +86 23 86730953. E-mail addresses: [email protected] (M.X. Deng), [email protected] (Y.X. Xiang). 1
2nd Revised manuscript submitted to Ultrasonics
1. Introduction Ultrasonic guided waves propagationg in layered piezoelectric plates have been applied in practical applications such as acoustic sensing, ultrasonic nondestructive evaluation, and signal processing, etc [1-3]. Generally, there are two sources of nonlinearity in primary guided wave propagation, namely, the convective nonlinearity independent of the material properties, and that produced by the nonlinear elastic, piezoelectric, and dielectric properties of the piezoelectric materials [4-6]. So, nonlinear effects will occur in conjunction with primary ultrasonic guided wave propagation in piezoelectric plates. One of the typical nonlinear effects is the generation of second harmonics. In recent years the studies on the effect of second-harmonic generation (SHG) by primary ultrasonic guided wave propagation attract more and more attention because of its potential to accurately assess the mechanical properties of plate-like structures [7-12]. However, the present studies on the SHG effect of ultrasonic guided waves are conducted only for the case where the materials of plate-like structures are assumed to be non-piezoelectric, and only the elastic nonlinearity of materials is taken into account [13-17]. It can be expected that besides the elastic nonlinearity in pizeoelectric materials, the piezoelectric and dielectric nonlinearities will likewise contribute to the SHG effect of primary guided wave propagation [5,6]. Moreover, besides the dispersion and multi-mode characteristics in guided wave propagation [13-17], the inherent coupling between the electric and mechanical fields in pizeoelectric materials will further increase the complexity of analyzing the SHG effect. Considering the potentials of using the SHG effect in practical applications [7-12], it is essential to analyze the SHG effect of primary ultrasonic 2
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guided wave propagation in piezoelectric plates, where the nonlinearities in elasticity, piezoelectricity, and dielectricity will be considered simultaneously. The aim of the present paper is to establish a model that can be used to analyze the said SHG effect. Especially, the analyses will focus on the influences of the electrical boundary conditions on the SHG effect of ultrasonic guided wave propagation in piezoelectric plates.
2. Theoretical fundamentals For simplicity, we just analyze ultrasonic guided wave propagation in a single piezoelectric plate. The piezoelectric material is assumed to be homogeneous with no attenuation and no dispersion. However, the present model can readily be extended for application to more complex piezoelectric plate-like structures. The Lagrangian coordinate system a1a2a3 established for a single piezoelectric plate is shown in Fig. 1, where γ1, γ2 and
γ3 are the angles of orientation between the two coordinate systems a1a2a3 and a1′a2′ a3′ , and the elastic, piezoelectric and dielectric constants of the piezoelectric material are originally defined under the coordinate system a1′a2′ a3′ . For a sound wave propagating along the Oa3 axis in Fig. 1, its formal solution is given by ui exp[j( ka3 + α ka2 − ω t )] under quasi-static approximation [18], where ui (i=1,2,3) corresponds to the mechanical displacement component along the Oai axis, and ui (i=4) the electrical potential. Based on the Christoffel equation for plane wave propagation in piezoelectric solids [18], the eight roots of α (denoted by α(q), q=1,…,8) can be obtained, and further, the corresponding eigensolution (u1( q ) , u2( q ) , u3( q ) ,ϕ ( q ) ) can also be determined for each
α(q). For notational simplicity, the Einstein summation convention is used throughout. 3
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Generally, a linear combination of eight eigensolutions (u1( q ) , u2( q ) , u3( q ) ,ϕ ( q ) ) constitutes the general solution of wave motion (neglecting the factor exp[-jωt]) [18] U i = Aqui( q ) exp[j( ka3 + α ( q ) ka2 )] , ϕ = Aqϕ ( q ) exp[j( ka3 + α ( q ) ka2 )],
(1)
where Ui (i=1,2,3) is the particle displacement component along the Oa1, Oa2 or Oa3 axis, and
ϕ the electrical potential, and Aq (q=1,…,8) are arbitrary constants to be determined. When a guided wave propagation is postulated, the corresponding dispersion equation can be obtained from the mechanical (stress-free at a2=0, h) and electrical (short-circuit or open circuit at a2=0, h) boundary conditions [18]. After that, the ratio of Aq to A1 can be determined by substituting the frequency-thickness product fh and the corresponding guided wave phase velocity into the equations of the mechanical and electrical boundary conditions [15]. The formal solution of a guided wave mode propagating in the piezoelectric plate (see Fig.1) has the same form as that shown in Eq. (1). Due to the anisotropy of the piezoelectric material, the guided wave propagation cannot generally be decoupled into two independent wave modes: Lamb wave and shear horizontal plate wave [15,18]. It should be noted here that the short-circuit boundary condition at the surface of piezoelectric plate (a2=0 or h in Fig. 1) means that the corresponding electrical potential at a2=0 or h is zero, while the open circuit boundary condition at the surface of piezoelectric plate (a2=0 or h) means that the corresponding surface charge density at a2=0 or h is zero [18]. Generally, the analysis of the SHG effect of primary guided wave propagation is extremely complicated due to its dispersion and multi-mode characteristics [13-17]. However, the elastic, piezoelectric and dielectric nonlinearities of piezoelectric materials are often small and the successive approximation approach can be applied [4-6]. Second-order perturbation 4
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reduces a complicated nonlinear problem to the linear one with known excitation sources, so that the linear analysis approach can be applied even if the original problem is nonlinear [13,14]. When the primary (f, p) guided wave (with the driving frequency f and the order p) propagates along the direction of the Oa3 axis in Fig. 1, in the interior of the solid plate, there are bulk driving force components with the frequency 2f, denoted by [4-6] fi ( 2 f ) = M ijklmn
∂ 2U k ∂U m ∂ 2ϕ ∂ϕ − d klij ∂a j ∂al ∂an ∂a j ∂ak ∂al
∂ 2U l ∂ϕ ∂U l ∂ 2ϕ + ( f kijlm + δ ilekmj ) + ∂a j ∂am ∂ak ∂am ∂a j ∂ak
,
(2)
where Mijklmn is defined by cijklmn+ δkmcijln+ δimcjnkl+δikcjlmn (i, j, k, l, m, n=1,2,3), and cijkl and cijklmn are second- and third-order elastic constants. The constant ekmj is second-order piezoelectric one. It should be noted that the electrostrictive constant dklij is related both to elasto-optic and electrostrictive effects, and that the third-order piezoelectric constant fkijlm is related to the electroacoustic effect [5,6]. Unlike wave motion in non-piezoelectric material, there is a bulk charge density with the frequency 2f, denoted by [5,6]
ρ ( 2 f ) = ε ijk
∂ 2ϕ ∂ϕ ∂ 2U j ∂U l + ( fijklm + δ jl eikm ) ∂ai∂a j ∂ak ∂ai ∂ak ∂am
∂ 2ϕ ∂U k ∂ϕ ∂ 2U k − d ijkl + ∂ai∂a j ∂al ∂a j ∂ai ∂al
,
(3)
where εijk is third-order dielectric constant. It is noted that all the material constants in Eqs. (2) and (3) should be transformed from the coordinate system a1′a2′ a3′ to a1a2 a3 [15,18,19]. Besides f i ( 2 f ) and ρ(2f) in the interior of the piezoelectric plate, there are second-order stress tensors (with the frequency 2f ) at the two surfaces of the piezoelectric plate, denoted by [4-6]
Pij( 2 f ) =
∂U k ∂U m 1 ∂ϕ ∂ϕ ∂U ∂ϕ 1 M ijklmn − d klij + ( f kijlm + δ il ekmj ) l , 2 ∂al ∂an 2 ∂ak ∂al ∂am ∂ak
and second-order surface charge density, denoted by [5,6] 5
(4)
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1 2
σ ( 2 f ) = ε 2 jk
∂ϕ ∂ϕ 1 ∂U ∂U l ∂ϕ ∂U k . + ( f 2 jklm + δ jl e2 km ) j − d 2 jkl ∂a j ∂ak 2 ∂ak ∂am ∂a j ∂al
(5)
The second-order terms, f i ( 2 f ) , ρ(2f), Pij( 2 f ) , and σ(2f), can provide a complete and accurate description for the SHG effect [5,6]. According to the modal analysis approach for waveguide excitation [13,14,18], these second-order terms can be regarded as excitation sources for the generation of a series of double frequency guided wave (DFGW) modes. Thus, the field of the SHG by the (f, p) guided wave propagation can formally be expressed by the summation of these DFGWs, i.e. [18], U i( 2) = α s (a3 ) ⋅ U i( 2 f ,s ) (a2 ) , ϕ ( 2) = α s (a3 ) ⋅ ϕ ( 2 f ,s ) (a2 ) ,
(6)
where U i( 2 f ,s ) (a2 ) (i=1,2,3) and ϕ(2f,s)(a2) are, respectively, the Oai-axis displacement component and the electrical potential of the DFGW mode (with the driving frequency 2f and the order s), and α s (a3 ) is the corresponding expansion coefficient. Based on the reciprocity relation and the orthogonality of guided waves, the equation governing αs(a3) is given by [13-18], FS + Fb ∂ ( 2 f ,s ) ∂a − j k α s (a3 ) = 4 P , 3 ss
(7)
where Pss is the average power flow per unit width along the Oa1 axis for the sth DFGW mode, and k(2f,s) is the Oa3-axis component of the wave vectors. The forcing function due to the bulk sources f i ( 2 f ) and ρ(2f) is given by [5,6,18] h ~ h Fb= 2 jω ∫0 [U i( 2 f ,s ) (a2 ) ⋅ f i ( 2 ) ] d a2 − j 2ω ∫0 [ϕ~ ( 2 f ,s ) (a2 ) ⋅ ρ ( 2) ] d a2 ,
(8)
and, after considerations of the mechanical and electrical boundary conditions at the two surfaces a2=0 and h, the forcing function due to the surface sources Pij( 2 f ) and σ(2f) is derived as follows [5,6,18] 6
2nd Revised manuscript submitted to Ultrasonics a2 = h ~ a2 = h FS = 2 jω[U i( 2 f , s ) (a2 ) ⋅ Pi 2( 2 f ) ] + 2 jω[ϕ~ ( 2 f ,s ) (a2 ) ⋅ σ ( 2 f ) ] a2 =0 ,
(9)
a2 = 0
where the sign ‘~’ on the top of U i( 2 f ,s ) and ϕ ( 2 f ,s ) denotes the complex conjugate operation. For the short-circuit or open circuit case at a2=0 or h, the second term in the right-hand side of Eq. (9) should, respectively, be removed or reserved. Considering an initial condition for a DFGW generation, i.e., αs(a3)=0 when a3=0, αs(a 3) can formally be given by [13,14]
α s (a3 ) =
α s sin[ D ka3 ] 4
D kh
exp[j k ( 2 f ,s ) a3 + j D ka3 ] ,
(10)
where αs=(FS+Fb)h exp(-2jka 3)/Pss; c(f,p) = ω/k and c(2f,s)=2ω/k(2f,s) are, respectively, the phase velocity of the (f, p) guided wave and the sth DFGW mode; the parameter D=[c(2f,s)-c(f,p)]/c(2f,s) can be used to describe the degree of dispersion (i.e., the relative difference of phase velocity). The solution given in Eq. (10) is exact within the second-order perturbation approximation. When αs ≠ 0, and the phase velocity matching condition between the primary mode and the DFGW [i.e., D=0 in Eq. (10)] is satisfied, αs(a3) can further be written as
αs(a3)=
α s a3 4
⋅
h
exp[j k ( 2 f ,s ) a3 ] ,
(11)
meaning that the sth DFGW component will increase with propagation distance. When αs ≠ 0 and D ≠ 0, there will be a beat effect for the amplitude of the sth DFGW with propagation distance, and the contribution of sth DFGW to U i( 2 f ) can be negligible [13]. These results are similar to that when a guided wave propagates in a non-piezoelectric plate [13-15]. Provided that the piezoelectric and dielectric nonlinearities of material are not considered (namely, the constants ekmj, dklij, and fkijlm equal zero), ρ(2) and σ(2f) in Eqs. (3) and (5) will vanish, and the mathematical expression of f i( 2 f ) and Pij( 2 f ) in Eqs. (2) and (4), as well as
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the solution of the DFGWs shown in Eqs. (8)-(10), will reduce to the same results as that found earlier in Refs. [13-15]. Thus, the analysis model proposed in this paper is more general than that derived previously. For a given set of orientation angles (γ1, γ2, γ3), the primary (f, p) guided wave propagating in the piezoelectric plate with the specific electrical boundary conditions at a2=0 and h (see Fig. 1) can be selectively generated to ensure αs ≠ 0 and D=0. In this way, the sth DFGW component will grow with propagation distance, and the obvious second-harmonic signals of primary guided wave propagation can be observed. When the changes in the electrical boundary conditions take place, the SHG effect of primary ultrasonic guided wave propagation will be influenced in the following two aspects. First, changes in the electrical boundary conditions will influence the dispersion relations of guided waves [18]. Thus, the original phase velocity matching condition D=0 [i.e., c(f,p)= c(2f,s)] may not be satisfied now when the electrical boundary conditions change. This will remarkably influence the efficiency of the SHG by the (f, p) guided wave propagation [13]. Second, changes in the electrical boundary conditions may provide different acoustic field features for the (f, p) guided wave. This will influence the magnitude of αs(a 3) because FS and Fb in Eq. (10) are proportional to the square of amplitude of the (f, p) guided wave [4-6].
3. Numerical analyses In the previous section, a theoretical model has been established for the SHG by primary ultrasonic guided wave propagation in a piezoelectric plate. Based on the model established, the possible influences of the electrical boundary conditions on the SHG have been 8
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qualitatively analyzed. Next some numerical analyses are given to understand the above theoretical analyses. The plate material is assumed to be single crystal LiNbO3 (Trigonal system, 3m), and its parameters are given in Refs. [19] and [20]. Under the open circuit conditions at a2=0 and h, the dispersion curves for the primary guided wave mode and the DFGWs propagating in the LiNbO3 plate with (γ1, γ2, γ3) = (0 o, 0o, 0o) are computed and shown in Fig. 2. The intersections between the vertical dotted line V (fh=4.75MHz.mm) and the dispersion curves of the DFGWs (denoted by points s0, s1, s2, s3 and s4, etc.) denote the DFGW mode components constituting the field of the second harmonic of the primary guided wave (denoted by point p 0) propagation. At this given frequency-thickness product fh=4.75MHz.mm, points p0 and s0 overlap, meaning that the primary guided wave (point p 0) and the DFGW (points s0) have the same phase velocity, i.e., c ( f , p0 ) = c ( 2 f , s0 ) = 7.364MHz.mm. Some parameters of the DFGWs (determined by points s0, s1, s2, s3 and s4, etc.) generated by the primary guided wave (point p0) propagation are listed in Table I. Fig. 3 shows the curves of amplitudes of these DFGW components (at the surface a2=h) versus propagation distance. Clearly, the amplitude of the DFGW (point s0) at the surface a 2=h grows linearly with propagation distance due to D=0 and
αs ≠ 0 in Eq. (10). For the other DFGW components determined by points s1, s2, s3 and s4, etc., their amplitudes exhibit a beat effect (oscillate) with propagation distance because of the phase velocity mismatching between the primary mode and the DFGW [i.e., D ≠ 0 in Eq. (10)] [13]. From the viewpoint of practical measurement, the DFGW component (point s0) plays a dominant role in the second-harmonic field of the primary guided wave mode (point p0), but 9
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the contribution of the other DFGW components (determined by points s1, s2, s3 and s4, etc.) to both U i( 2) and ϕ ( 2) in Eq. (6) can be neglected after the primary guided wave has propagated some distance [13]. So it can be regarded that the field of the SHG is only determined by the DFGW mode (point s0). Next, the analyses will focus on the influences of the electrical boundary conditions on the generation of the sth DFGW. Generally speaking, changes in the electrical boundary conditions will lead to changes in the phase velocity of guided waves, as well as FS, Fb, and
αs in Eqs. (7) and (10). When the orientation angles (γ1, γ2, γ3)= (0o, 0o, 0o) are kept unchanged, the phase velocity curves versus fh under different electrical boundary conditions are shown in Fig. 4. When the electrical boundary conditions change from the 1 st case (open circuit at a2=0 and h) to the 2nd case (open circuit at a2=0 and short-circuit at a2= h ) or the 3 rd case (short-circuit at a2=0 and h ), the mode pair of the primary and DFGW modes (denoted by intersections between the vertical dotted line V and the dispersion curves in Fig. 4) changes from point pair (p0, s0) to ( p ' , s ' ) or ( p ' ' , s ' ' ). Clearly, points p ' and s ' , as well as p ' ' and s' ' in Fig. 4, no longer overlap, meaning that there is a phase velocity mismatching between
the primary mode and the DFGW (i.e., D ≠ 0). When the frequency-thickness product fh is given by the vertical dotted line V (fh =4.75MHz.mm), some parameters of the sth DFGW (denoted, respectively, by point s0, s ' or s ' ' ) generated by the primary guided wave propagation (denoted, respectively, by point p0, p ' or p ' ' ) are listed in Table II. When the electrical boundary conditions change (relative to the open circuit conditions at a 2=0 and h), the amplitudes of the sth DFGW (at the surface a 2=h ) versus propagation distance are shown in Fig. 5. Obviously, the numerical results reveal that, using different 10
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electrical boundary conditions, the SHG effect of the primary guided wave propagation can be effectively regulated, relative to a specific electrical boundary condition that can ensure αs ≠ 0 and D=0 in Eq. (10). Specifically speaking, at the propagation distance given by the vertical dotted line H in Fig. 5, the SHG efficiency (characterized by the sth DFGW amplitude that is determined by the y-axis value of point D0, D1, or D2 ) can be effectively modified by changing the electrical boundary conditions.
4. Conclusions The nonlinearity in ultrasonic guided wave propagation is treated as a second-order perturbation to the linear wave motion response. Based on the modal expansion analysis for waveguide excitation, an accurate description for the SHG effect of primary ultrasonic guided wave propagation in a piezoelectric plate has been presented within a second-order perturbation approximation. The formal solution of the DFGWs, constituting the field of second harmonic, has been developed. The analytical results clearly reveal that the SHG effect of primary guided wave propagation is closely related to the electric boundary conditions of the piezoelectric plate. It is found that under different electrical boundary conditions there is an evident difference in the SHG effect of ultrasonic guided waves, and that the SHG effect is highly sensitive to the electrical boundary conditions. The results obtained may provide a means for regulating the SHG efficiency of ultrasonic guided waves by changing the electrical boundary conditions of piezoelectric plates.
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Acknowledgement The authors are grateful for the support provided by the National Natural Science Foundations of China under Grant Nos. 11274388, 11474093 and 11474361.
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References [1] M.H. Badi, G.G. Yaralioglu, A.S. Ergun, S.T. Hansen, E.J. Wong, B.T. Khuri-Yakub, Capacitive micromachined ultrasonic Lamb wave transducers using rectangular membranes, IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 50 (2003) 1191-1203. [2] J.L. Rose, A baseline and vision of ultrasonic guided wave inspection potential, Journal of Press Vessel Technology, 124 (2002) 273-282. [3] A.H. Nayfeh, Wave Propagation in Layered Anisotropic Media, Elsevier, Amsterdam, 1995. [4] A.N. Norris, in Nonlinear Acoustics, edited by M.F. Hamilton and D.T. Blackstock, Academic, New York, 1998. [5] D.H. McMahon, Acoustic second-harmonic generation in piezoelectric crystals, J. Acoust. Soc. Am. 44 (1968) 1007-1013. [6] V.E. Ljamov, Nonlinear acoustical parameters of pizeoelectric crystals, J. Acoust. Soc. Am. 52 (1972) 199-202. [7] M.X. Deng, Characterization of surface properties of a solid plate using nonlinear Lamb wave approach, Ultrasonics, 44 (2006) e1157-e1162. [8] M.X Deng, J.F. Pei, Assessment of accumulated fatigue damage in solid plates using nonlinear Lamb wave approach, Appl. Phys. Lett. 90 (2007) 121902. [9] C. Bermes, J.Y. Kim, J.M. Qu, L.J. Jacobs, Experimental characterization of material nonlinearity using Lamb waves, Appl. Phys. Lett. 90 (2007) 021901. [10] C. Pruell, J.Y. Kim, J.M. Qu and L.J. Jacobs, Evaluation of fatigue damage using nonlinear guided waves, Smart Mater. Struct. 18 (2009) 035003.
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[11] Y.X. Xiang, M.X. Deng, F. Xuan, Cumulative second-harmonic analysis of ultrasonic Lamb waves for ageing behavior study of modified-HP austenite steel, Ultrasonics, 51(2011) 974-981. [12]. W.B. Li, Y. Cho, J.D. Achenbach, Detection of thermal fatigue in composites by second harmonic Lamb waves, Smart Mater. Struct. 21 (2012) 085019. [13] M.X. Deng, Analysis of second-harmonic generation of Lamb modes using a modal analysis approach, J. Appl. Phys. 94 (2003) 4152-4159. [14] W.J.N. de Lima, M.F. Hamilton, Finite-amplitude waves in isotropic elastic plates, J. Sound. Vib. 265 (2003) 819–839. [15] M.X. Deng, Second-harmonic generation of ultrasonic guided wave propagation in an anisotropic solid plate, Appl. Phys. Lett. 92 (2008) 111910. [16] V.K. Chillara, C.J. Lissenden, Interaction of guided wave modes in isotropic weakly nonlinear elastic plates: higher harmonic generation, Journal of Applied Physics 111 (2012) 124909. [17] V.K. Chillara, C.J. Lissenden, Analysis of second harmonic guided waves in pipes using a large-radius asymptotic approximation for axis-symmetric longitudinal modes, Ultrasonics, 53 (2013) 862-869. [18] B.A. Auld, Acoustic Fields and Waves in Solids, Vols. I and II, Wiley, New York, 1973. [19] D.F. Nelson, Electric, Optic, and Acoustic Interactions in Dielectrics, Wiley, New York, 1979. [20] Y. Cho, K. Yamanouchi, Nonlinear, elastic, piezoelectric, electrostrictive, and dielectric constants of lithium niobate, J. Appl. Phys. 61 (1987) 875-887.
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Table I. Some parameters associated with the SHG at fh=4.75MHz.mm under open circuit conditions.
a
Primary mode
( f , p0 )
( f , p0 )
( f , p0 )
( f , p0 )
( f , p0 )
DFGW
( 2 f , s0 )
(2 f , s1 )
( 2 f , s2 )
( 2 f , s3 )
( 2 f , s4 )
c(f, p ) (MHz.mm)
7.364
7.364
7.364
7.364
7.364
c(2f, s) (MHz.mm)
7.364
7.086
5.078
4.024
9.479
αs a
(16.92, 0.09)
(-0.02, -2.10) (2.32, 0.83) (-1.07, 12.15) (-14.02. 0.65)
The unit or scale of α sU i( 2 f ,s ) (a2 ) (i=1,2,3) is Am2 h , where Am is the amplitude of the
Oa 3-axis displacement component of the (f, p) guided wave propagation at the surface a 2=0.
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Table II. Some parameters associated with the SHG at fh=4.75MHz.mm under different electrical boundary conditions.
a
Primary mode
( f , p0 )
( f , p' )
( f , p' ' )
DFGW
( 2 f , s0 )
(2 f , s ' )
(2 f , s ' ' )
c(f, p) (MHz.mm)
7.364
7.311
7.243
c(2f, s) (MHz.mm)
7.364
7.338
7.296
αs a
(16.92, 0.09)
(23.13, -0.96)
(38.53, -4.00)
The unit or scale of α sU i( 2 f ,s ) (a2 ) (i=1,2,3) is Am2 h , where Am is the amplitude of the
Oa 3-axis displacement component of the (f, p) guided wave propagation at the surface a 2=0.
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Figure Captions
Fig. 1. (Color online) The piezoelectric plate and the Lagrangian coordinate system.
Fig. 2. (Color online) Phase velocity dispersion curves for the primary guided wave and the DFGWs under the open circuit conditions.
Fig. 3. (Color online) Amplitudes of the sth DFGW vs propagation distance under the open circuit conditions: (a) the Oa 2-axis displacement component, and (b) the Oa3-axis displacement component. The Oa 1-axis displacement component vanishes for the case of (γ1,
γ2, γ3) = (0o, 0o, 0o).18
Fig. 4. (Color online) Phase velocity dispersion curves for the primary guided waves and the DFGWs under different electrical boundary conditions.
Fig. 5. (Color online) Amplitudes of the sth DFGW vs propagation distance for different electrical boundary conditions: (a) the Oa2-axis displacement component, and (b) the Oa 3-axis displacement component. The Oa 1-axis displacement component vanishes for the case of (γ1,
γ2, γ3) = (0o, 0o, 0o).18
17
Figure 1
Figure 2
Figure 3(a)
Figure 3(b)
Figure 4
Figure 5(a)
Figure 5(b)
2nd Revised manuscript submitted to Ultrasonics
This paper establishes a model that can be used to analyze the second-harmonic generation (SHG) of primary ultrasonic guided wave propagation in piezoelectric plates. Previous studies are conducted only for the case where the materials of plate-like structures are assumed to be non-piezoelectric, and only the elastic nonlinearity of materials is taken into account. The present work not only takes into account of the effect of elastic nonlinearity of materials, but also considers the nonlinearities in elasticity, piezoelectricity, and dielectricity simultaneously. It has been found out that the electrical boundary conditions of piezoelectric plates have a large influence on the SHG effect of ultrasonic guided wave propagation. 1. A model has been established for analyzing the SHG effect of ultrasonic guided waves in piezoelectric plates. 2. The possible influences of electrical boundary conditions of piezoelectric plates on the SHG effect have been analyzed. 3. The results obtained provide a means through which the SHG efficiency of ultrasonic guided wave propagation can be effectively regulated by changing the electrical boundary conditions of piezoelectric plates.
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