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Subharmonic Generation in Piezoelectrics with Cantor-Like Structure C. CHIROIU, V. CHIROIU, L. MUNTEANU, C. RUGINA Romanian Academy, Institute of Solid Mechanics,Ctin Mille 15, 70701 Bucharest, Romania, E-mail: chiroiu@mecsol.ro E. RUFFINO , M. SCALERANDI Politecnico di Torino, Dipartimento di Fisica,Corso Duca degli Abruzzi 24, 10129 Torino, Italy, E-mail: scalerandi@polito.it Contact

Abstract

The purpose of this paper is to explain, by the soliton theory, the subharmonic generation in piezoelectric plates with Cantor-like structure (PCS). Craciun et al. [1] and Alippi et al. [2-4] have shown the experimental evidence of extremely low thresholds for subharmonic generation of ultrasonic waves in one-dimensional PCS, as compared to the corresponding homogeneous and periodical structures. The origin of this apparent anomaly was theoretically investigated by the same scientists by studying anharmonic coupling between normal modes.

1. Introduction

Craciun et al. [1] and Allipi et al. [2-4] show experimental evidence of extremely low thresholds for subharmonic generation of ultrasonic waves (SWG) in one-dimensional artificial piezoelectric plates with Cantor-like structure, as compared to the corresponding homogeneous and periodical plates. An anharmonic coupling between the extended-vibration (phonon) and the localized-mode (fracton) regimes explained this phenomenon. They demonstrate that the large enhancement of non-linear interaction results from the more favorable frequency and spatial matching of coupled modes (fractons and phonons) in the Cantor-like structure.
In this paper we solve the nonlinear equations, which govern the SWG phenomenon by a generalization of Fourier series, which uses the cnoidal wave as the fundamental basis function [8,9,10]. The cnoidal waves are much richer than sine waves, i. e. the modulus m of the cnoidal wave (0£m£1) can be varied to obtain a sine wave (m@0), Stokes wave (m@0.5) or soliton (m@1).
We provide theoretically the existence of multiple fracton and multiple phonon-mode regimes in the displacement field for a piezoelectric plate with Cantor-like structure. The nonlinear wave motion is a linear superposition of cnoidal waves plus additional terms, which include nonlinear interactions among the waves. The nonlinear interactions among the cnoidal waves are significant for the explanation the draining of the energy away from the input wave towards low frequencies spectrum.

2. Formulation of the problem

We consider a composite plate formed by alternating elements of nonlinear isotropic piezoelectric ceramics (PZ) and epoxy resin (ER), following a triadic Cantor sequence. Craciun et al.[1] and Alippi et al.[2] constructed an artificial one-dimensional Cantor structure. We consider the same sample using a triadic Cantor sequence up to the fourth generation (31 elements). The origin of the coordinate system Ox₁x₂x₃ is located at the left end, in the middle plane of the sample, with the axis Ox₁ in-plane and normal to the layers and Ox₃ out-plane, normal to the plate. The length of the plate is l, the width of the smallest layer is l/18 and the thickness of the plate is h. Let the regions occupied by the plate be where V^p and V^e are the regions occupied by PZ and ER layers. The boundary surface of V be S partitioned in the following way
          where
is the boundary surface of V^p,
is the boundary surface of V^e , and

Let the unit outward normal of S be n_i the interfaces between constituents be I^pe.
The governing equations are composed from:
1. The quasistatic motion equations for V^p [4]

   (2.1)

   (2.2)

   where r_p is the density, u_iis the displacement vector, t_ij is the stress tensor, D_iis the electric induction vector, E_i is the electric field and i is the electric potential.

   The constitutive equations of nonlinear isotropic piezoelectrics are [5]

   (2.3)

   (2.4)

   (2.5)

   where e_ij is the strain tensor, l^p,m^p are the Lame constants, A^p,B^p,C^p are the Landau constants, are the linear and nonlinear dielectric constants, , and are the linear and nonlinear coefficients of piezoelectricity and . We consider all quantities are independent with respect to x₂ and u₂ = 0, E₂ = 0. We have

   (2.6)

   (2.7)

   We express the elastic potentials f and y

   (2.8)

   in the form

   (2.9)

   where and are the unknown functions.
2. The boundary conditions on

   (2.10)

   (2.11)

   where , are quantities prescribed on the boundary and is the Maxwell stress tensor. We consider that a periodical electric field is applied to the both surfaces of the plate to excite the Lamb waves, over a wide frequency range (10kHz < w/2p < 5MHz).
   The action of this field is described by Maxwell stress tensor [6,8]

   (2.12)

3. The motion equations for V^e

   (2.15)

   (2.16)

   The field of displacement u₁, u₃ is expressed in the form (2.8) with

   (2.17)

   with the same unknown functions and as in (2.9).

4. The boundary conditions on

   (2.18)

5. The boundary conditions on S₂

   (2.16)

6. The conditions on interfaces between constituents I^pe (the displacement and the traction vectors are continuous)

   (2.17)

   where the bracket indicates a jump across the interface and k = 1,3.

3. Solutions

We have observed that the governing equations (2.1)-(2.2) and (2.15) can support for and certain particular functions and may be written in terms of the q- function representation [9,10,13]

(3.1)

where x º x₁ and

(3.2)

Here n is the finite number of degrees of freedom for a particular solution of the problem. The n phases are given by

(3.3)

where k_j are the wave numbers, w_j the frequencies and b_j the phases.
The parameters k_j, w_j, b_j and B_ij, as well as the unknowns parameters A,B,C,D,E,F can be determined from the eigenvalue problem obtained by substituting (3.1) in the motion equations [13].

4. Results and conclusions

The calculus was carried out for l= 67.5 mm and h= 0.3 mm. The material constants are shown in table 1.

	piezoelectric ceramics	epoxy resin
l	71.6 Gpa	42.31 GPa
m	35.8 Gpa	3.76 GPa
A	-2000 Gpa	2.8 GPa
B	-1134 Gpa	9.7 GPa
C	-900 Gpa	-5.7 Gpa
	4.065 nF/m	-
	2.079 nF/m	-
e1	-0.218 nm/V	-
	-0.435 nm/V	-
r	7650 Kg/m3	1170Kg/m3
Table 1: The material constants for piezoelectric ceramics and epoxy resin

Table 2 shows the computed frequencies and the errors obtained at the optimal solution given by the genetic algorithm.

wn / 2p	100.2 ± 0.05	167 ± 0.01	217.1 ± 0.03	250.5 ± 0.1	334 ± 0.01	367.4 ± 0.01	417.5 ± 0.1	501 ± 0.02	584.5 ± 0.03
	617.9 ± 0.01	668 ± 0.03	835 ± 0.06	935.2 ± 0.06	1085.5 ± 0.1	1169 ± 0.07	1269.2 ± 0.02	1503 ± 0.05	1670 ± 0.4
	1770.2 ± 0.2	1987.3 ± 0.12	2120.9 ± 0.02	2250 ± 0.1	2471.6 ± 0.3	2655.3 ± 0.01	2672 ± 0.01	2972.6 ± 0.2	3340 ± 0.4
	3690.7 ± 0.01	3774.2 ± 0.15	3991.3 ± 0.24	4250 ± 0.03	4291.9 ± 0.06	4525.7 ± 0.2	4826.3 ± 0.01
Table 2: Estimation results: computed eigenfrequencies

Resonant vibration modes are excited by applying an external electric field on both sides of the plate with w = w_n. The undetermined coefficients P are approximatelly determined from a genetic algorithm. In fig. 1 the admittance curve (k / rw vs w / 2p) (fig.1) in the linear regime ( @ 0.1V) marks by peaks the frequencies w = w_n of the modes. If is increases above a threshold value = 5.27 V the w / 2 p subharmonic generation is observed. The amplitude of waves is calculated at the surface of the plate as a function of . Figs.2-4 show the displacements of the normal modes w / 2 p = 334 kHz, 501 kHz, 835 kHz and respectively of the subharmonic modes w / 4p=167 kHz, 250.5 kHz, 417.5 kHz. Two kind of vibration regimes are found: a localised-mode (fracton) regime represented in fig.5 for w / 2p=1169 kHz, 2672 kHz and 3340 kHz and an extended-vibration (phonon) regime represented in fig. 6 for w / 2p= 4175 kHz and 4250 kHz. A sketch of the plate geometry is given on the abscissa (dashed, piezoelectric ceramic and white, epoxy resin The fracton vibrations are mostly localised on a few elements, while the phonon vibrations essentially extend to the whole plate. In the case of a periodical plate the dispersion prevents good frequency matching between the fundamental and appropriate subharmonic modes. For the homogeneous plate the mismatch w_n - w/2 is due to the symmetry of fundamental modes with respect to x. Only symmetric odd n can induce a subharmonic, but never w/2 coincides with a plate vibration mode. For a Cantor plate, we have obtained the same result as Craciun and Alippi [1.2]: given a normal mode w_n, for excitation at w = w_n, the value of the expected threshold E_th i. e. the ability of generating the w/2 subharmonic, is determined by the existence of a normal mode with: (i) small frequency mismatch w_n - w/2, and, (ii) large spatial overlap between the fundamental and subharmonic displacement field.

Fig 1: The admittance-frequency curve for the Cantor plate

Fig 2: The amplitudes of the surface displacement of the normal mode w /2p= 334 kHz and of the subharmonic mode w /4p= 167 kHz.

Fig 3: The amplitudes of the surface displacement of the normal mode w /2p= 501 kHz and of the subharmonic mode w /4p= 250.5 kHz.

Fig 4: The amplitudes of the surface displacement of the normal mode w /2p= 835 kHz and of the subharmonic mode w /4p= 417.5 kHz.

Fig 5: The normal amplitudes for two localised vibration modes (w /2p=1169 kHz, w /2p=2672 kHz and w /2p=3340 kHz)

Fig 6: The normal amplitudes for two localised vibration modes (w /2p=4175 kHz and w /2p=4250 kHz.

Acknowledgments.

Support for this work by The National Agency for Science, Technology and Innovation (ANSTI ) Bucharest, Grant nr.5208/99-B3 and B8, is gratefully acknowledged.

References

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