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Abstract

The vibration modes of an elastic plate are usually divided into propagating and non-propagating (evanescent) kinds. Non-propagating wave modes are very important for guided wave inspection of defect size and shape. But it is difficult to obtain the complex solutions of the transcendental dispersion equation, corresponding to the non-propagating wave. In this article, we present an improved Legendre polynomial method to calculate the complex-valued dispersion and study properties of the non-propagating wave in a piezoelectric spherical plate. Comparisons with other related studies are conducted to validate the correctness of the presented method. The complete dispersion and attenuation curves are plotted in three-dimensional frequency-complex wave number space. The influences of material piezoelectricity and radius–thickness ratio on non-propagating waves in piezoelectric spherical plates are investigated. The amplitude distributions of the electric potential and displacement are also discussed in detail. All the results presented in this work can provide theoretical guidance for ultrasonic nondestructive evaluation and are promising to be applied to improve the resolution of piezoelectric transducers.

Keywords

Non-propagating piezoelectric spherical plate Legendre polynomial method complex dispersion attenuation radius–thickness ratio

Introduction

Due to the unique properties of coupling between mechanical and electrical properties, piezoelectric materials are widely used in several industry fields, such as vibration control devices, electromechanical transducers, measuring instruments, actuators, and filters.¹ The research of waves propagating in piezoelectric materials has been a topic of particular importance and interest. Complete dispersion relations and wave mode shapes are essential for in-depth understanding the dynamic behaviors of piezoelectric structures, which is useful to design and optimize the high-precision transducers and sensors.²

Many computational models and methods have been developed to investigate wave propagation in various piezoelectric structures in the past few decades. By the transfer matrix method, Stewart and Yong³ investigated the propagating wave characteristics in anisotropic piezoelectric multilayer plates. Using the reverberation-ray matrix approach, Guo et al.⁴ studied the guided wave propagation in multilayered piezoelectric plates. Based on the three-dimensional (3D) piezoelectric theory, Syngellakis and Lee⁵ analyzed wave propagation in an infinite piezoelectric plate. Many curved structures, such as cylindrical and spherical structures, are extensively used in all sorts of engineering fields. Wave propagations in such structures have got much attention, although their treatment is more complex, difficult, and challenging than that of the plane structures. Using the Fourier transform, Paul and Venkatesan⁶ determined the axial wave solutions in a piezoelectric hollow cylinder. Using the wave potentials method, Puzyrev and Storozhev⁷ investigated the problem of wave propagation in piezoelectric hollow cylinders of sector cross section and presented the dispersion spectra. Yu et al.⁸ investigated the guided propagating wave in piezoelectric spherical plates using polynomial approach. Dai and Wang⁹ presented an analytical method to investigate the wave propagating in piezoelectric laminated spherical plates based on the Laplace transforms and the finite Hankel transforms. As is reviewed above, many research concerns are paid on propagating waves, but research on non-propagating waves is relatively limited so far.

Recently, research on pseudo surface acoustic waves (PSAW) in piezoelectric half-spaces reveals that the PSAW modes have higher velocities and lower attenuations than the classical surface acoustic waves.^10,11 Such modes make the piezoelectric device possess higher resolution. Similarly, non-propagating waves also have these features. According to classification method of the wave mode by Auld,¹² the complete guided wave includes propagating wave modes and non-propagating wave modes at a certain frequency. The wave number of propagating wave mode is purely real. Propagating wave is very useful for location or for detection of defects in the structure. Non-propagating wave modes are fundamentally different from propagating wave modes, because they decay with the propagation distance. The non-propagating wave with purely imaginary wave number exponentially decays with the propagation distance and exists only near the edges, loads, and defects. The non-propagating wave with complex wave number is another kind of non-propagating wave, a damped sinusoidal attenuation rather than an exponential attenuation, which is very important for guided wave inspection of defect size and shape.¹³ In the field of theory and application of guided waves, many problems, such as structural analysis of acoustic sensors and resonant devices, will involve the solution of the dispersion equation in the complex-valued wave number domain. However, there is one more unknown variable for solving complex roots than for solving purely real roots and solving purely imaginary roots, which makes the computational difficulty increase sharply. In large number of papers involving non-propagating wave, only the purely imaginary solutions are given. An¹⁴ studied the evanescent Lamb waves in the plates and revealed the advantages of identification and location of cracks by the evanescent waves. Using the spectral collocation method, Quintanilla et al.¹⁵ studied the propagation and evanescent waves in plates and cylinders and gave a full 3D dispersion curve. Chen et al.¹⁶ theoretically studied the shear horizontal (SH) waves in a piezoelectric plate of cubic crystals and presented the full spectrum. Based on a semi-analytical approach, Bai et al.¹⁷ presented the full spectra for a homogeneous piezoelectric cylinder. Yan and Yuan^18,19 presented a quantitative study of conversion of evanescent Lamb waves into propagating in isotropic plates, and they also investigated the conversion of evanescent SH waves into propagating using a semi-analytical approach. Recently, Cheng et al.²⁰ investigated the complex dispersion relations and the evanescent wave modes in periodic beams using the extended differential quadrature element method. Dubuc et al.²¹ presented a numerical approach based on spectral methods for the computation of complete wave modes in nonuniformly stressed plates. The existing research on non-propagating wave is limited to the simple materials or simple structures such as elastic materials and flat structures, but the study on piezoelectric materials and spherical curved structures is very rare.

In this article, we improved the Legendre polynomial method to compute the complex dispersion relation and investigate the properties of non-propagating guided waves in a piezoelectric spherical plate. The presented method can obtain the complete solutions by transforming the complicated wave equations into a simple eigenvalue problem. One known case is given to validate the method. Properties of non-propagating guided waves in various piezoelectric spherical plates are investigated, and the complete 3D dispersion and attenuation curves are plotted to gain a more in-depth insight into the nature of the non-propagating waves. The influences of material piezoelectricity and radius–thickness ratio on the non-propagating waves are investigated. Moreover, the amplitude distributions of the physical quantities including the electric potential and displacement are illustrated.

Mathematics and formulation of the problem

Based on the theory of 3D piezoelectric elasticity, a piezoelectric spherical curved plate with free surfaces inside and outside is considered in spherical coordinate system Oθφr. And a is the inner radius, b the outer radius, h the plate thickness, and η = b/h denotes the radius–thickness ratio.

The system of governing equations involves the motion equations, the electrical displacement equilibrium equations and the constitutive equations

\begin{matrix} \frac{\partial σ_{rr}}{\partial r} + \frac{1}{r} \frac{\partial σ_{r θ}}{\partial θ} + \frac{1}{r \sin θ} \frac{\partial σ_{r φ}}{\partial φ} + \frac{2 σ_{rr} + σ_{r θ} \cot θ - σ_{θ θ} - σ_{φ φ}}{r} = ρ \frac{\partial^{2} u_{r}}{\partial t^{2}} \\ \frac{\partial σ_{r θ}}{\partial r} + \frac{1}{r} \frac{\partial σ_{θ θ}}{\partial θ} + \frac{1}{r \sin θ} \frac{\partial σ_{θ φ}}{\partial φ} + \frac{3 σ_{r θ} + \cot θ (σ_{θ θ} - σ_{φ φ})}{r} = ρ \frac{\partial^{2} u_{θ}}{\partial t^{2}} \\ \frac{\partial σ_{r φ}}{\partial r} + \frac{1}{r} \frac{\partial σ_{θ φ}}{\partial θ} + \frac{1}{r \sin θ} \frac{\partial σ_{φ φ}}{\partial φ} + \frac{3 σ_{r φ} + 2 σ_{θ φ} \cot θ}{r} = ρ \frac{\partial^{2} u_{φ}}{\partial t^{2}} \\ \frac{\partial D_{r}}{\partial r} + \frac{1}{r} \frac{\partial D_{θ}}{\partial θ} + \frac{1}{r \sin θ} \frac{\partial D_{φ}}{\partial φ} + \frac{2 D_{r} + D_{θ} \cot θ}{r} = 0 \end{matrix}

(1)

σ_{ij} = C_{ijkl} ε_{kl} - e_{kij} E_{k}

(2a)

D_{j} = e_{jkl} ε_{kl} + \in_{jk} E_{k}

(2b)

where σ_ij, ε_ij, D_i, and E_i denote the stress, strain, electrical displacement and electrical intensity components, respectively. u_i denotes the mechanical displacement component in the ith (i = r, θ, φ) direction. C_ijkl, e_kij, $\in_{ik}$ are the elastic, piezoelectric, and the dielectric constants, respectively. ρ is the mass density. Superposed dot and comma in subscripts denote time and spatial derivatives, respectively.

The relationship between the mechanical displacement and strain, and the relationship between the electrical intensity and electric potential on the basis of the quasi-static Maxwell’s equation are, respectively

\begin{matrix} ε_{rr} = \frac{\partial u_{r}}{\partial r}, ε_{θ θ} = \frac{1}{r} \frac{\partial u_{θ}}{\partial θ} + \frac{u_{r}}{r}, ε_{φ φ} = \frac{1}{r \sin θ} \frac{\partial u_{φ}}{\partial φ} + \frac{u_{r}}{r} + \frac{\cot θ}{r} u_{θ} \\ ε_{r θ} = \frac{1}{2} (\frac{1}{r} \frac{\partial u_{r}}{\partial θ} + \frac{\partial u_{θ}}{\partial r} - \frac{u_{θ}}{r}), ε_{r φ} = \frac{1}{2 r} (\frac{1}{\sin θ} \frac{\partial u_{r}}{\partial φ} - u_{φ}) + \frac{1}{2} \frac{\partial u_{φ}}{\partial r} \\ ε_{θ φ} = \frac{1}{2 r} (\frac{1}{\sin θ} \frac{\partial u_{θ}}{\partial φ} + \frac{\partial u_{φ}}{\partial θ} - u_{φ} \cot θ), E_{r} = - \frac{\partial Φ}{\partial r}, E_{θ} = - \frac{1}{r} \frac{\partial Φ}{\partial θ}, E_{φ} = - \frac{1}{r \sin θ} \frac{\partial Φ}{\partial φ} \end{matrix}

(3)

where Φ is the electric potential.

The traction-free and electricity open-circuit boundary conditions require that $σ_{rr} |_{r = a, b} = 0$ , $σ_{r θ} |_{r = a, b} = 0$ , $σ_{r φ} |_{r = a, b} = 0$ , and $D_{r} |_{r = a, b} = 0$ . Considering the boundary condition, the material parameters depending on the position can be expressed as

\begin{matrix} C = C \cdot f (r), ρ = ρ \cdot f (r), e = e \cdot f (r), \in = \in \cdot f (r) \\ f (r) = {\begin{matrix} \begin{matrix} 1, & a \leq r \leq b \end{matrix} \\ \begin{matrix} 0, & elsewhere \end{matrix} \end{matrix} \end{matrix}

(4)

where f(r) is a rectangular window function, and its derivative is $δ (r - a) - δ (r - b)$ . Then, the above-mentioned boundary conditions can be automatically incorporated in the constitutive equations.²²

For the wave propagating in the circumferential direction of a piezoelectric spherical waveguide, the electric potential and the mechanical displacement can be written as

u_{i} = \exp (ikb φ - i ω t) U_{i} (r), Φ = \exp (ikb φ - i ω t) X (r)

(5)

where U_i and X represent the vibration amplitude in the ith (i = r, θ, φ) direction and the amplitude of electric potential, respectively. k and $ω$ respectively denote wave number and angular frequency, and i is the imaginary number.

We substitute equations (3)–(5) into equation (2) with following substitution into equation (1). Consequently, the governing differential equations about the electric potential and displacement components can be obtained. Here, the case of an orthotropic, piezoelectric spherical plate with radial polarization is given

\begin{matrix} [r^{2} (C_{33} U ″ + e_{33} X ″) + 2 r C_{33} U' + i rkb (C_{23} + C_{44}) W' + r (2 e_{33} - e_{31} - e_{32}) X' + (C_{23} + C_{13} - C_{22} \\ - 2 C_{12} - C_{11} - k^{2} b^{2} C_{44}) U] + i kb (C_{23} - C_{44} - C_{22} - C_{12} -) W - k^{2} b^{2} e_{24} X] f (r) \\ + f' (r) [r^{2} (C_{33} U' + e_{33} X') + r [(C_{13} + C_{13}) U + i kb C_{23} W]] = - ρ r^{2} ω^{2} Uf (r) \end{matrix}

(6a)

[r^{2} C_{55} V ″ + 2 r C_{55} V' - (C_{12} + 2 C_{55} + k^{2} b^{2} C_{66}) V] f (r) + f' (r) (r^{2} C_{55} V' + r C_{55} V) = - ρ r^{2} ω^{2} Vf (r)

(6b)

\begin{matrix} [r^{2} C_{44} W ″ + i rkb (C_{44} + C_{23}) U' + 2 r (ikb C_{24} + C_{44}) W' + i rkb (e_{32} + e_{24}) X' + i kb (2 C_{44} + C_{12} + C_{44} + C_{12} +) U \\ + (C_{66} - 2 C_{44} - k^{2} b^{2} C_{22}) W + (2 i kb e_{24} - k^{2} b^{2} e_{22}) X] f (r) + f' (r) (r^{2} C_{44} W' + i rkb C_{44} U \\ - r C_{44} W + i kb e_{24} X) = - ρ r^{2} ω^{2} Wf (r) \end{matrix}

(6c)

\begin{matrix} [r^{2} (e_{33} U ″ - \in_{33} X ″) + r (2 e_{33} + e_{32} + e_{31}) U' - 2 r \in_{33} X' + irkb (e_{24} + e_{32}) W' + (e_{31} + e_{32} - k^{2} b^{2} e_{24}) U \\ - k^{2} b^{2} \in_{22} X + ikb (e_{32} - e_{24}) W] f (r) + f' (r) [ikb e_{32} W - r^{2} (\in_{33} X' - e_{33} U') + r (e_{31} + e_{32}) U] = 0 \end{matrix}

(6d)

where U, V, and W respectively represent the vibration amplitude in the radial and two tangential (θ and φ) directions. The subscript comma indicates the partial derivative with respect to the radius r.

Obviously, equations (6a), (6c), and (6d) are mutually coupled. They are associated with Lamb-like wave. Equation (6b) is independent since the material is orthotropic, which represents SH wave and can be solved analytically. Because equation (6b) is easily solved, we here just give the solving process of Lamb-like wave as below.

We expand the field quantities into the Legendre polynomial form as^23,24

\begin{matrix} U (r) = \sum_{m = 0}^{\infty} p_{m}^{1} Q_{m} (r), W (r) \\ = \sum_{m = 0}^{\infty} p_{m}^{2} Q_{m} (r), X (r) = \sum_{m = 0}^{\infty} p_{m}^{3} Q_{m} (r) \end{matrix}

(7)

where $p_{m}^{i}$ represents the expansion coefficient, $Q_{m} (r) = \sqrt{(2 m + 1) / h} P_{m} (2 r - (b + a) / h)$ , with $P_{m}$ being the mth order polynomial.

Substituting equation (7) into equations (6a), (6c), and (6d), then multiplying both sides of the modified equations (6a), (6c), and (6d) by a complex conjugate $Q_{j}^{*} (r)$ , then integrating over r in the interval [a, b], and exploiting orthonormality of Legendre polynomials, yields

\begin{matrix} k^{2} [\begin{matrix} A_{11}^{j, m} & A_{12}^{j, m} & A_{13}^{j, m} \\ A_{21}^{j, m} & A_{22}^{j, m} & A_{23}^{j, m} \\ A_{31}^{j, m} & A_{32}^{j, m} & A_{33}^{j, m} \end{matrix}] {\begin{matrix} p_{m}^{1} \\ p_{m}^{2} \\ p_{m}^{3} \end{matrix}} \\ + k [\begin{matrix} B_{11}^{j, m} & B_{12}^{j, m} & B_{13}^{j, m} \\ B_{21}^{j, m} & B_{22}^{j, m} & B_{23}^{j, m} \\ B_{31}^{j, m} & B_{32}^{j, m} & B_{33}^{j, m} \end{matrix}] {\begin{matrix} p_{m}^{1} \\ p_{m}^{2} \\ p_{m}^{3} \end{matrix}} \\ + [\begin{matrix} C_{11}^{j, m} & C_{12}^{j, m} & C_{13}^{j, m} \\ C_{21}^{j, m} & C_{22}^{j, m} & C_{23}^{j, m} \\ C_{31}^{j, m} & C_{32}^{j, m} & C_{33}^{j, m} \end{matrix}] {\begin{matrix} p_{m}^{1} \\ p_{m}^{2} \\ p_{m}^{3} \end{matrix}} \\ = - ω^{2} [\begin{matrix} M_{m}^{j} & 0 & 0 \\ 0 & M_{m}^{j} & 0 \\ 0 & 0 & 0 \end{matrix}] {\begin{matrix} p_{m}^{1} \\ p_{m}^{2} \\ p_{m}^{3} \end{matrix}} \end{matrix}

(8)

where $M_{m}^{j}$ , $A_{α β}^{j, m}, B_{α β}^{j, m}, and C_{α β}^{j, m} (α, β = 1, 2, 3)$ are the matrix elements, and they can be deduced from equation (6) and are presented in Appendix 1.

Equation (8) can be abbreviated as

k^{2} A \cdot p + k^{1} B \cdot p + C \cdot p = - ω^{2} H \cdot p

(9)

where A, B, C, and H are matrices of order 3m × 3m, $p = [p_{m}^{1} p_{m}^{2} p_{m}^{3}]^{T}$ .

Equation (9) is a positive-definite eigenvalue problem with real roots ω². If our interest is in the propagating wave, as the previous research work,⁸ it is very efficient to specify real k and then solve for ω. But for the non-propagating wave, the simple approach is not useful because wave number k is complex, thus leading to a multivariable search. In order to overcome this difficulty, we develop a new solution procedure as shown below.

Introducing a new column vector

q = k \cdot p

(10)

That is, $q = [q_{m}^{1} q_{m}^{2} q_{m}^{3}]^{T} = k [p_{m}^{1} p_{m}^{2} p_{m}^{3}]^{T}$ .

Then, equation (9) can be written as

k \cdot A \cdot q + B \cdot q = - (C - H) p

(11)

Multiplying the two sides of equation (11) by the inverse matrix A⁻¹, and rearranging the terms yields

A^{- 1} (H - C) p - (A^{- 1} B) q = k \cdot q

(12)

Combining equations (10) and (16) and defining $R = [p q]^{T}$ , we have

[\begin{matrix} 0 & I \\ A^{- 1} (H - C) & - A^{- 1} B \end{matrix}] R = k \cdot R

(13)

where 0 represents a zero matrix and I is the identity matrix.

Up to this stage, the problem is transformed into a generalized eigenvalue solving problem. Equation (13) is easily solved to use the Mathematica function “Eigenvalues” for the problem of the type AX = λX, which can yield the complex wave number eigenvalues k(ω).

Numerical results

Approach and computer program validation

To the best of our knowledge, non-propagating wave in a piezoelectric spherical plate has not been presented before. In order to validate the presented approach and the computer program, we compute a piezoelectric spherical plate of large radius–thickness ratio, which may be equal to a plate. We compare our results with the existing ones of a piezoelectric plate from the reverberation-ray matrix method. The material is PZT-4 and the corresponding parameters are given in Table 1. The non-dimensional frequency $Ω = ω h / (2 π \sqrt{C_{55} / ρ})$ and the non-dimensional wave number γ = kh/(2π) are introduced. Re(γ) denotes the real part of the complex wave number γ, and Im(γ) the imaginary part. The numerical results are shown in Figure 1, where the red dotted lines are our results and the black ones are the literature results from the reverberation-ray matrix method.⁴ It can be seen from Figure 1 that the calculated results are very consistent.

Table 1.

Material parameters of PZT-4.

C ₁₁	C ₂₂	C ₃₃	C ₁₂	C ₁₃	C ₂₃	C ₄₄	C ₅₅	C ₆₆
13.9	13.9	11.5	7.78	7.43	7.43	2.56	2.56	3.06
e ₁₅	e ₂₄	e ₃₁	e ₃₂	e ₃₃	ϵ₁₁	ϵ₂₂	ϵ₃₃	ρ
12.7	12.7	−5.2	−5.2	15.1	646.4	646.4	562.2	7.5

Units: C_ij (×10¹⁰ N/m²), e_ij (C/m), ϵ_ij (×10⁻¹¹ F/m²), ρ (×10³ kg/m³).

Figure 1.

Dispersion curves: black dotted lines from the reverberation-ray matrix method and red dotted lines from the present method.

3D dispersion curves

Propagating guided wave has got a lot of attention, and here we put the emphasis on non-propagating guided waves in piezoelectric spherical structures. To have a better understanding of the nature of the non-propagating wave, a full 3D dispersion curve is plotted. The material is Ba₂NaNb₅O₁₅ and the corresponding parameters are given in Table 2.

Table 2.

Material parameters of Ba₂NaNb₅O₁₅.

C ₁₁	C ₂₂	C ₃₃	C ₁₂	C ₁₃	C ₂₃	C ₄₄	C ₅₅	C ₆₆
23.9	24.7	13.5	10.4	5	5.2	6.5	6.6	7.6
e ₁₅	e ₂₄	e ₃₁	e ₃₂	e ₃₃	ϵ₁₁	ϵ₂₂	ϵ₃₃	ρ
2.8	3.4	−0.4	−0.3	4.3	196	201	28	5.3

Units: C_ij (×10¹⁰ N/m²), e_ij (C/m), ϵ_ij (×10⁻¹¹ F/m²), ρ (×10³ kg/m³).

The 3D complete dispersion curves of SH wave propagation in the piezoelectric spherical plate with η = 10 are given in Figure 2. For clarity, different colored dotted lines represent different modes. The dark blue ones are purely real solutions and green ones purely imaginary solutions. Since all of wave numbers corresponding to a given frequency is calculated at each time, the dispersion curves consist of a series of continuous Re(γ)–Im(γ)–Ω levels “slices.” The points that make up the dispersion curve become sparse when become close to the cutoff frequencies on the Ω-axis. We can see from Figure 2 that SH non-propagating wave has only the purely imaginary mode. These solutions including the real and imaginary always come out in pairs of opposite signs, which means that different wave fields are propagated in positive and negative directions. These non-propagating wave modes begin at the ω = 0 plane and terminate at a certain cutoff with increasing frequency. But SH propagating wave modes infinitely increase from the cutoff with increasing wave number. Interestingly, a purely imaginary mode transforms into a propagating wave mode at cutoff.

Figure 2.

3D full spectrum of SH wave: dark blue dotted lines—purely real solutions; green dotted lines—purely imaginary solutions.

The 3D full spectrum of Lamb-like wave is given in Figure 3. Dark blue and green lines represent the same meaning with the above description, and red ones represent complex solutions. Figure 3 shows that Lamb-like non-propagating wave has purely imaginary modes and complex modes. For purely imaginary branches, some modes with small imaginary wave numbers usually begin at a certain cutoff and terminate the adjacent one, and form many arcs. Most modes with large imaginary wave numbers (greater than 0.2) usually begin at ω = 0 plane and terminate at cutoffs of the higher-order propagation modes, and their dispersion curves become approximately vertical lines at low frequency. Also with the increase of frequency, the value of the imaginary wave number gradually decreases and the curves become approximately horizontal lines. For complex modes, most modes begin at ω = 0 plane and collapse onto the local extremes of the real branches. This means that the complex branches convert into the propagating. With increasing frequency, some complex branches connecting two adjacent purely imaginary branches at high frequency appear, as pointed out in Quintanilla et al.¹⁵ These complex branches start from the local maximum of one purely imaginary branch to the local minimum of another one. These curves indicate effectively the function of local extremes as potential wells that trap the complex branches. The properties of local extremes are not very clear at present. It was shown by Onoe et al.²⁵ that the location of local extremes is very sensitive to Poisson’s ratio and structural geometry. Some of the special complex branches mentioned above will have different characteristics with different Poisson’s ratios or structures of cross-section geometries, and some non-propagating wave modes will not even appear for different structures. Moreover, the spectra of Lamb-like wave are also symmetrical, and the conjugate of the complex wave number is also the solution of the equation.

Figure 3.

Dispersion of Lamb-like wave: dark blue dotted lines—purely real solutions; green dotted lines—purely imaginary solutions; red dotted lines—complex solutions.

The phase velocity dispersion and attenuation curves of the first several propagating and complex modes of Lamb-like wave are shown in Figure 4. Dimensionless phase velocity and attenuation and frequency are respectively adopted as $Vp = ω / (Re (k) \cdot \sqrt{C_{55} / ρ})$ , Im(kh) and $fh = ω h / (2 π \sqrt{C_{55} / ρ})$ . These curves reveal that the phase velocity of propagating modes decreases and gradually tends to a stable value with increasing frequency, but that of non-propagating modes with complex wave number increases with increasing frequency. The phase velocity of the non-propagating (complex wave number) mode is much bigger than that of the propagating mode at high frequency. For instance, the phase velocity of the fifth complex mode is more than 6 but that of the propagating modes is below 2, for fh = 3–4. Also the wave dispersion is very weak for this non-propagating mode at high frequency.

Figure 4.

Phase velocity dispersion and attenuation curves of Lamb-like wave, (a) and (b) are from different viewpoints; propagating wave in blue; non-propagating wave in red.

Influences of radius–thickness ratios and material piezoelectricity on dispersion

Considering the symmetry of the spectrum, only one quadrant is presented for clarity and comparison. Figure 5 shows the 3D spectrum for the piezoelectric spherical plate with a small radius–thickness ratio η = 2. Comparing Figures 2 and 3 with Figure 5, we can find that the wave number of SH non-propagating wave decreases with the decrease of the radius–thickness ratio, which means SH non-propagating wave decays more slowly for a piezoelectric spherical plate with a small radius–thickness ratio. For Lamb-like non-propagating waves, the real wave number of the complex branch is usually small and also becomes smaller with the decrease of the radius–thickness ratio. That means the propagation velocity of the non-propagating wave is decreased with the increase of the radius–thickness ratio. In contrast to the case of the large radius–thickness ratio, for the plate with a small radius–thickness ratio, the first inflection point of the purely real branch appears in the second-order mode.

Figure 5.

3D spectrum for the piezoelectric spherical plate with η = 2: (a) Lamb-like wave and (b) SH wave.

Next, we investigate the influences of the material piezoelectricity on the dispersion. Assuming that piezoelectric and dielectric coefficients are zero and elastic parameters and mass density keep unchanged. The corresponding 3D spectrum for the spherical plate with η = 10 is shown in Figure 6, with the same color scheme as shown in Figure 3. Comparing with Figure 3, we can notice that the piezoelectricity has an evident influence on the propagating and non-propagating wave. Those purely imaginary branches beginning at 0 frequency and ending at the cutoff and those complex branches interconnecting two imaginary branches disappear (in the given range).

Figure 6.

3D spectrum of Lamb-like wave for the elastic spherical plate.

Displacement and electric potential distributions

To reveal the characteristics of non-propagating waves, we calculate the distributions of the displacement and electric potential for a piezoelectric spherical plate with η = 10, which can be obtained by equations (5) and (7). Here we consider a special position (marked with a circle in Figure 3) where the complex branch first terminates at the minimal value of the fourth propagating mode, Ω, is about 0.7. The corresponding imaginary part of the complex wave number is very small. Figures 7 and 8 present the amplitude distributions of the displacement and electric potential in thickness direction and wave propagation direction, when Ω = 0.73248 and the corresponding complex wave number γ = 0.25894–0.04301i, and when Ω = 0.74045 and the corresponding real wave number γ = 0.23807, and when Ω = 0.73248 and the corresponding imaginary wave number γ = –0.28879i, respectively. We can note that the distributions are nearly symmetrical. Figures 7(a) and 8(a) indicate that the complex-valued modes can propagate a quite far distance (a few tens of the plate thickness) and show a damped sinusoid attenuation. Comparing Figure 7(a) with Figure 7(b), and Figure 8(a) with Figure 8(b), we can find that the corresponding displacement u_φ and electric potential distribution are almost identical, which indicates the non-propagating mode becomes propagating with increasing frequency. Figures 7(c) and 8(c) show that the non-propagating wave mode with purely imaginary wave number exponentially decays and does not propagate with propagation distance.

Figure 7.

Displacement distributions: (a) complex wave number mode, (b) purely real wave number mode, and (c) purely imaginary wave number mode.

Figure 8.

Electric potential distributions: (a) complex wave number mode, (b) purely real wave number mode, and (c) purely imaginary wave number mode.

Conclusion

An improved Legendre polynomial approach is successfully developed to investigate the non-propagating waves in a piezoelectric spherical plate. This approach enables one to determine the complex dispersion relations and to obtain the full spectrum. The correctness of this approach is verified by numerical results. Properties of non-propagating waves in various piezoelectric spherical plates are investigated. From the numerical results, some conclusions can be drawn as follows:

The presented approach can transform the coupled differential wave equations into an eigenvalue problem and can find all eigenvalues without iteration, including the real, imaginary, and complex root of the transcendental dispersion equation.

SH non-propagating wave has only the purely imaginary modes, and Lamb-like non-propagating wave has purely imaginary and complex modes. For complex modes, local inflection points occasionally occur on high-order real branches, and they turn into the propagating mode with increasing frequency.

The phase velocity of non-propagating (complex wave number) mode is much bigger than that of the propagating mode at high frequency, and the wave dispersion is very weak for the complex mode in some frequency ranges.

The influence of radius–thickness ratio and material piezoelectricity on the non-propagating wave is evident. The decay of the non-propagating wave becomes slower with reducing the radius–thickness ratio.

The purely imaginary-valued mode decays exponentially and does not propagate with propagation distance. But the complex-valued modes can propagate a quite far distance and show a damped sinusoid attenuation.

Footnotes

Appendix 1

\begin{matrix} A_{11}^{j, m} = - b^{2} C_{44} u (m, 0, 0, j), A_{13}^{j, m} = - b^{2} e_{24} u (m, 0, 0, j), A_{22}^{j, m} = - b^{2} C_{22} u (m, 0, 0, j), \\ A_{31}^{j, m} = - b^{2} e_{24} u (m, 0, 0, j), A_{33}^{j, m} = b^{2} \in_{22} u (m, 0, 0, j), A_{12}^{j, m} = A_{21}^{j, m} = 0, A_{23}^{j, m} = A_{32}^{j, m} = 0; \\ B_{12}^{j, m} = ib (C_{23} + C_{44}) u (m, 1, 1, j) + ib (C_{23} - C_{12} - C_{22} - C_{44}) u (m, 0, 0, j) + ib C_{23} k (m, 1, 0, j), \\ B_{21}^{j, m} = ib (C_{23} + C_{44}) u (m, 1, 1, j) + ib (2 C_{44} + C_{12} + C_{22}) u (m, 0, 0, j) + ib C_{44} k (m, 1, 0, j), \\ B_{23}^{j, m} = ib (e_{32} + e_{24}) u (m, 1, 1, j) + 2 ib e_{24} u (m, 0, 0, j) + ib e_{24} k (m, 1, 0, j), \\ B_{32}^{j, m} = ib (e_{32} + e_{24}) u (m, 1, 1, j) + ib (e_{32} - e_{24}) u (m, 0, 0, j) + ib e_{32} k (m, 1, 0, j), \\ B_{11}^{j, m} = B_{22}^{j, m} = B_{33}^{j, m} = 0, B_{13}^{j, m} = B_{31}^{j, m} = 0; \\ C_{11}^{j, m} = C_{33} u (m, 2, 2, j) + 2 C_{33} u (m, 1, 1, j) + (C_{13} + C_{23} - C_{11} - C_{22} - 2 C_{12}) u (m, 0, 0, j) \\ + C_{33} k (m, 2, 1, j) + (C_{13} + C_{23}) k (m, 1, 0, j), \\ C_{13}^{j, m} = e_{33} u (m, 2, 2, j) + (2 e_{33} - e_{31} - e_{32}) u (m, 1, 1, j) + e_{33} k (m, 2, 1, j), \\ C_{22}^{j, m} = C_{44} u (m, 2, 2, j) + 2 C_{44} u (m, 1, 1, j) + (C_{66} - 2 C_{44}) u (m, 0, 0, j) + C_{44} k (m, 2, 1, j) - C_{44} k (m, 1, 0, j), \\ C_{31}^{j, m} = e_{33} u (m, 2, 2, j) + (e_{31} + e_{32} + 2 e_{33}) u (m, 1, 1, j) + (e_{31} + e_{32}) u (m, 0, 0, j) \\ + e_{33} k (m, 2, 1, j) + (e_{31} + e_{32}) k (m, 1, 0, j), \\ C_{33}^{j, m} = - \in_{33} u (m, 2, 2, j) - 2 \in_{33} u (m, 1, 1, j) - \in_{33} k (m, 2, 1, j), \\ C_{12}^{j, m} = C_{21}^{j, m} = 0, C_{23}^{j, m} = C_{32}^{j, m} = 0; \\ M_{m}^{j} = ρ u (m, 2, 0, j) \end{matrix}

where

u (m, l, n, j) = \int_{a}^{b} Q_{j}^{*} (r) r^{l} \frac{\partial^{n} Q_{m} (r)}{\partial r^{n}} dr; k (m, l, n, j) = \int_{a}^{b} Q_{j}^{*} (r) r^{l} \frac{\partial X (r)}{\partial r} \frac{\partial^{n} Q_{m} (r)}{\partial r^{n}} dr

Handling Editor: Muhammad Akhtar

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work was supported by the National Natural Science Foundation of China (no. U1504106), the fundamental research funds for the national outstanding youth project of Henan Polytechnic University (no. NSFRF140301), and the Program for Innovative Research Team of Henan Polytechnic University (T2017-3).

ORCID iD

Xiaoming Zhang

References

Chang

. Analysis of wave propagation in infinite piezoelectric plates. J Mech 2005; 21: 103–108.

Chen

Guo

Pan

EN.

Wave propagation in magneto-electro-elastic multilayered plates with nonlocal effect. J Sound Vib 2017; 400: 550–563.

Stewart

Yong

YK.

Exact analysis of the propagation of acoustic waves in multilayered anisotropic piezoelectric plates. IEEE T Ultrason Ferr 1994; 41: 375–390.

Guo

Chen

Zhang

YL.

Guided wave propagation in multilayered piezoelectric structures. Sci China Ser G: Phys Mech Astro 2009; 52: 1094–1104.

Syngellakis

Lee

PCY

. Piezoelectric wave dispersion curves for infinite anisotropic plates. J Appl Phys 1993; 73: 7152–7161.

Paul

Venkatesan

Wave propagation in a piezoelectric solid cylinder of arbitrary cross section. J Acoust Soc Am 1987; 82: 2013–2020.

Puzyrev

Storozhev

Wave propagation in axially polarized piezoelectric hollow cylinders of sector cross section. J Sound Vib 2011; 330: 4508–4518.

Lefebvre

Guo

YQ.

Wave propagation in multilayered piezoelectric spherical plates. Acta Mechanica 2013; 224: 1335–1349.

Dai

Wang

Stress wave propagation in laminated piezoelectric spherical shells under thermal shock and electric excitation. Eur J Mech A: Solid 2005; 24: 263–276.

10.

Glushkov

Glushkova

Zhang

CZ.

Surface and pseudo-surface acoustic waves piezoelectrically excited in diamond-based structures. J Appl Phys 2012; 112: 064911.

11.

Roshchupkin

Ortega

Snigirev

et al . X-ray imaging of the surface acoustic wave propagation in La₃Ga₅SiO₁₄ crystal. Appl Phys Lett 2013; 103: 154101.

12.

Auld

BA.

Acoustic fields and waves in solids. 2nd ed. Malabar, FL: Krieger Publishing Company, 1990.

13.

Huang

Dong

SB.

Propagating waves and edge vibrations in anisotropic composite cylinders. J Sound Vib 1984; 96: 363–379.

14.

YK.

Measurement of crack-induced non-propagating Lamb wave modes under varying crack widths. Int J Solid Struct 2015; 62: 134–143.

15.

Quintanilla

Lowe

MJS

Craster

. Full 3D dispersion curve solutions for guided waves in generally anisotropic media. J Sound Vib 2016; 363: 545–559.

16.

Chen

Wang

et al . Propagation of shear-horizontal waves in piezoelectric plates of cubic crystals. Arch Appl Mech 2016; 86: 517–528.

17.

Bai

Taciroglu

Dong

et al . Elastodynamic Green’s functions for a laminated piezoelectric cylinder. Int J Solid Struct 2004; 41: 6335–6350.

18.

Yan

Yuan

FG.

Conversion of evanescent Lamb waves into propagating waves via a narrow aperture edge. J Acoust Soc Am 2015; 137: 3523–3533.

19.

Yan

Yuan

FG.

A semi-analytical approach for SH guided wave mode conversion from evanescent into propagating. Ultrasonics 2018; 84: 430–437.

20.

Cheng

Shi

YL.

Complex dispersion relations and evanescent waves in periodic beams via the extended differential quadrature method. Compos Struct 2018; 187: 122–136.

21.

Dubuc

Ebrahimkhanlou

Salamone

Computation of propagating and non-propagating guided modes in nonuniformly stressed plates using spectral methods. J Acoust Soc Am 2018; 143: 3220–3230.

22.

Datta

Hunsinger

BJ.

Analysis of surface waves using orthogonal functions. J Appl Phys 1978; 49: 475–479.

23.

Matar

Gasmi

Zhou

et al . Legendre and Laguerre polynomial approach for modeling of wave propagation in layered magneto-electro-elastic media. J Acoust Soc Am 2013; 133: 1415–1424.

24.

Othmani

Takali

Njeh

Modeling of phase velocity and frequency spectrum of guided Lamb waves in piezoelectric-semiconductor multilayered structures made of AlAs and GaAs. Superlattice Microst 2017; 111: 396–404.

25.

Onoe

Mcniven

Mindlin

RD.

Dispersion of axially symmetric waves in elastic rods. J Appl Mech 1962; 29: 729–734.

Complex dispersion solutions for guided waves and properties of non-propagating waves in a piezoelectric spherical plate

Abstract

Keywords

Introduction

Mathematics and formulation of the problem

Numerical results

Approach and computer program validation

3D dispersion curves

Influences of radius–thickness ratios and material piezoelectricity on dispersion

Displacement and electric potential distributions

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

ORCID iD

References