Analysis of Axisymmetric and Non-Axisymmetric Wave Propagation in a Homogeneous Piezoelectric Solid Circular Cylinder of Transversely Isotropic Material

Ultrasonic non-destructive evaluation relies on thorough understanding of the propagating and evanescent waves in the material under investigation. This study could be useful in such applications as, for example, the identification of defects of finite size across the circumference of a piezoelectric rod, since reflections of waves from such defects are generally non-axisymmetric in nature (Rose, 1999). Also, results obtained in this paper can serve as a basis for design of piezoelectric transducers, for which standing wave modes resulting from reflections of travelling waves at cross-section boundaries of the cylinder play an important role. The broad development of the finite (FEM) and boundary element methods (BEM) also needs some reference results obtained from exact solutions. In this case the proposed investigation of the non-axisymmetric propagating and evanescent waves in a piezoelectric cylinder could help to create reliable test beds of FEM and BEM. The study of wave propagation in systems with cylindrical geometry has been undertaken by a number of investigators. While the systems considered so far have been both isotropic and anisotropic in nature, a relatively larger literature exists for isotropic materials. The first investigator of waves propagating in a solid isotropic cylinder was Pochhammer (Pochhammer, 1876). Other investigators of the subject can be found in standard texts (Achenbach, 1984), (Graff, 1991), and (Rose, 1999). For cylinders composed of anisotropic materials Chree (Chree, 1890) investigated propagation of the axisymmetric waves. Much later Mirsky (Mirsky, 1964) investigated a problem of non-axisymmetric wave propagation in transversely isotropic circular solid and hollow cylinders. Other contributors to the subject were (Armenakas & Reitz, 1973), (Frazer, 1980), (Nayfeh & Nagy (1995), (Berliner & Solecki, 1996), (Niklasson & Datta, 1998), and (Honarvar, et al., 2007), just to name a few. Numerical results for the dispersion of axisymmetric guided waves in a composite cylinder with a transversely isotropic core were presented by (Xu & Datta, 1991). Source: Wave Propagation in Materials for Modern Applications, Book edited by: Andrey Petrin, ISBN 978-953-7619-65-7, pp. 526, January 2010, INTECH, Croatia, downloaded from SCIYO.COM


Introduction
Ultrasonic non-destructive evaluation relies on thorough understanding of the propagating and evanescent waves in the material under investigation. This study could be useful in such applications as, for example, the identification of defects of finite size across the circumference of a piezoelectric rod, since reflections of waves from such defects are generally non-axisymmetric in nature (Rose, 1999). Also, results obtained in this paper can serve as a basis for design of piezoelectric transducers, for which standing wave modes resulting from reflections of travelling waves at cross-section boundaries of the cylinder play an important role. The broad development of the finite (FEM) and boundary element methods (BEM) also needs some reference results obtained from exact solutions. In this case the proposed investigation of the non-axisymmetric propagating and evanescent waves in a piezoelectric cylinder could help to create reliable test beds of FEM and BEM. The study of wave propagation in systems with cylindrical geometry has been undertaken by a number of investigators. While the systems considered so far have been both isotropic and anisotropic in nature, a relatively larger literature exists for isotropic materials. The first investigator of waves propagating in a solid isotropic cylinder was Pochhammer (Pochhammer, 1876). Other investigators of the subject can be found in standard texts (Achenbach, 1984), (Graff, 1991), and (Rose, 1999). For cylinders composed of anisotropic materials Chree (Chree, 1890) investigated propagation of the axisymmetric waves. Much later Mirsky (Mirsky, 1964) investigated a problem of non-axisymmetric wave propagation in transversely isotropic circular solid and hollow cylinders. Other contributors to the subject were (Armenakas & Reitz, 1973), (Frazer, 1980), (Nayfeh & Nagy (1995), (Berliner & Solecki, 1996), (Niklasson & Datta, 1998), and (Honarvar, et al., 2007), just to name a few. Numerical results for the dispersion of axisymmetric guided waves in a composite cylinder with a transversely isotropic core were presented by (Xu & Datta, 1991). For piezoelectric cylinders the analysis of vibrations of circular shells was performed by Paul (Paul, 1966). Investigation of axisymmetric waves in layered piezoelectric rods with open circuit electric field conditions and their composites was carried out in (Nayfeh et al., 2000). The axisymmetric problem of the wave propagation in a piezoelectric transversely isotropic rod was analysed for rigid sliding and elastic simply supported mechanical boundary conditions and different types of electric field conditions in (Wei & Su, 2005). Several papers were devoted to development of numerical and finite element methods of investigation of piezoelectric cylinders. In (Siao et al., 1994) the authors solved the problem of wave propagation in a laminated piezoelectric cylinder via the FEM. Their paper contains tables and graphs of dispersion curves for real and imaginary values of wavenumbers. Unfortunately the data contains a misprint in a scale factor for the dimensionless wave number, making it quite difficult to compare their results with the results of the present paper. This misprint was further corrected in (Bai et al., 2004), who in their paper studied the electromechanical response of a laminated piezoelectric hollow cylinder by means of a semi-analytical FEM formulation. In (Shatalov & Loveday, 2004), and (Bai et al., 2006) the authors also studied the phenomenon of end reflections of waves in a semi-infinite layered piezoelectric cylinder. Our approach parallels that of (Mirsky, 1964) and (Berliner & Solecki, 1996). It is also similar to (Winkel et al., 1995) who analytically solved the problem of wave propagations in an infinite cylindrical piezoelectric core rod immersed into an infinite piezoelectric cladding material. In (Winkel et al., 1995) the authors focused on the pure guided waves with real solutions of the determining bi-cubic equation. It was found in the process of investigation that this approach is applicable to the general problem of propagating and evanescent waves with real and complex solutions. In our approach the three dimensional equations of elastodynamics together with the quasi-electrostatic Gauss law are solved in terms of seven displacements and three electric potentials, each satisfying the Helmholtz equations. In contrast to (Winkel et al., 1995) we propose a simple approach to solution of the problem which has an additional advantage that in the limiting case of small electric and electromechanical constants our results automatically coincide with the classical results of (Mirsky, 1964) and (Berliner & Solecki, 1996) for a transversely isotropic cylinder. This evidence gives one confidence in the correctness of the obtained results. Imposing the allowed mechanical and electric boundary conditions on the cylinder surface, the characteristic or dispersion equation is derived in the form of a determinant of the fourth order in the non-axisymmetric case and a determinant of the third order in the axisymmetric case. On the basis of the obtained results a numerical algorithm for displaying the dispersion curves is developed. The algorithm is based on calculation of the logarithm of modulus of the left hand side of the dispersion equation on a discrete mesh in the "wavenumberfrequency" ( k ω − ) plane. If the left hand side value of the dispersion equation tends to zero the logarithm tends to minus infinity. This method produces sharp negative spikes on the surface plot and automatically provides a picture of configuration of the dispersion curves. Similar approach of simultaneous qualitative displaying of the dispersion equation solutions was used in (Honarvar et al., 2006) that produced a 3-D cross-section of the real part of left hand side of the equation. The main advantage of our approach is that the local minima of the logarithm of modulus of the left hand side of the dispersion equation's matrix give proper approximations of the roots which can be further used as guess values of solutions of the dispersion equation. In the present chapter we present the dispersion curves of solid www.intechopen.com Analysis of Axisymmetric and Non-Axisymmetric Wave Propagation in a Homogeneous Piezoelectric Solid Circular Cylinder of Transversely Isotropic Material 337 cylinders made from two piezoelectric materials -PZT-4 and PZT-7A for open-and shortcircuit electric boundary conditions and for different circumferential wavenumbers m = 1, 2, 3 and for the axisymmetric case. The curves demonstrate substantial influence of the electric boundary conditions and electromechanical coupling on behaviour of the dispersion curves. These results obtained from exact solution of the problem could serve as the reference data and help researchers to create reliable FEM techniques for analysis of vibration of piezoelectric bodies.

Formulation of problem
In this section we establish the equations of motion and boundary conditions, and obtain solution of the problem within the framework of the following assumptions and approximations: linear elasticity, linear constitutional model of piezoelectricity, quasi-static approximation of the electric field, axial polarization of the piezoelectric material, neglecting of thermal effects, absence of free charges in the material, and boundary and body forces. Axis Oz coincides with the axis of the cylinder, ,, rz θ -radius, polar angle and axial coordinate, and ,, uvw-radial, tangential and axial displacements correspondingly. Equations of motion and boundary conditions of the cylindrical piezoelectric body are derived from the Hamilton variational principle. In order to apply Hamilton's principle we define the Lagrangian of the system as follows: Where K is the kinetic energy, U is the strain energy and W is the electric energy of the system:  Possible boundary conditions on the outer cylindrical surface obtained from the Hamilton principle are as follows: (cylindrical surface is free in the radial direction) or (8a) (cylindrical surface is free in the tangential direction) or (9a) www.intechopen.com The Euler-Lagrange equation could be also presented in the standard form of the Navier equations of the linear dynamical elasticity (Redwood, 1960) Electric boundary conditions will be considered in both open-and close-circuit form (11a) or (11b) and one of these conditions will be added to boundary conditions (13).
In the explicit form the system (9) or (12) of equations is as follows: ( ) The explicit form of boundary conitions (13) is:

Analytic solution of the problem
We seek the solution of the problem (14) -(15) in terms of harmonic travelling waves along z -axis in terms of several displacement and electric potentials first introduced by (Mirsky, 1964) and further used by (Winkel et al 1995) and (Shatalov et al., 2009) as follows: After substitution (16) into (14) we obtain the following system of equations:   (18)  Second equation of system (18) must be compatible with the third and fourth equations of system (17), i.e. we have the following system of equations: System (22) has nontrivial solutions if the determinant of the system is zero: www.intechopen.com where coefficients η j , μ j are obtained from system of equations (22) after substitution expressions (29): Finally the solution of system (14) using representation (16) is as follows: where m is a positive integer in the non-axisymmetric case,   (Berliner & Solecki, 1996) and hence, all the presented results are converted into the well known results (Mirsky, 1964) and (Berliner & Solecki, 1996) which also coincides with the known result (Berliner & Solecki, 1996). Hence, the results obtained contain the classical results of investigation of a passive transversely isotropic material as a particular limiting case.

Axisymmetric case
Axisymmetric vibrations of the piezoelectric cylinder could be considered as a particular case of the general problem with 0 m = , 0 v = , 2 0 E = , 46 0 TT = = and all variables are θindependent. In this case system of equations (14) is rewritten as follows: ( )

Numerical results and discussion
In this section we present the dispersion curves for non-axisymmetric waves with circumferential wavenumbers 1,2,3 m = and axisymmetric waves resulting from the characteristic equation (38). Two piezoelectric materials are chosen, PZT-4 and PZT-7A, to illustrate the influence of electro-mechanical coupling coefficients on the configuration of the dispersion curves. The relevant material parameters for PZT-4 and PZT-7A are given in the Table 1. In order to obtain the dispersion curves we made use of a method similar to the novel method (Honarvar et al., 2008) where the dispersion curves are not produced as a result of solving of the dispersion equation by a traditional iterative find-root algorithm but are obtained by a zero-level cut in the velocity-frequency plane. In this chapter we modify this approach calculating the logarithm of modulus of determinant (37)  ω . Another advantage of this method is that it is much faster than the traditional root finding methods and as fast as the Honarvar method (Honarvar et al., 2008). The main disadvantage of this approach is that the roots of characteristic arguments are also displayed on the surface plots as obvious artefacts. An elaborate discussion of these artefacts is given by Yenwong-Fai, (Yenwong-Fai, 2008). These artefacts could be simply detected and eliminated from the dispersion plots by program tools.Our algorithm, as it has been implemented, does not search for branches of the dispersion relation well away from the real and imaginary axes for k. It would be relatively straightforward in principle to locate these additional branches.
Dispersion curves of bending waves ( ) ) pixels. The same resolution is used for Fig. 2 -17. The first dispersion curve of the propagating waves (real values of the wavenumber) tends to an asymptote of the surface wave propagation. It is joined to the second curve which tends to the asymptote of the shear waves through the domain of the evanescent waves. In Fig. 4 and 5 the dispersion curves of PZT-7A material are presented for the open-and short-circuit lateral surfaces respectively. In comparison with PZT-4 this material has lower values of the electro-mechanical coupling coefficients but practically the same elastic coefficients and mass density.
www.intechopen.com It follows from Fig. 1 -5 that the first fundamental mode of the bending waves is not sensitive to the nature of the electric boundary conditions on the lateral cylindrical surface. Furthermore it is practically not sensitive to the measure of electro-mechanical coupling of the material. The higher order modes are more sensitive to the nature of the electric boundary condition as well as to the measure of the electro-mechanical cross-coupling. Fig.  3 -5 shows that dispersion curves differ quite substantially from the curves in Fig. 1 and 2. This difference is explained mainly by the factor that the electro-mechanical coupling coefficients of PZT-7A are less than the corresponding factors of PZT-4. It is reflected in undulating behaviour of the propagating higher modes as well as the values of the cut-off frequencies. The PZT-4 cylinder with short-circuit lateral surface demonstrates a negative slope of the fourth branch in a quite broad range of wavenumbers (Fig. 1). Substantial dependence of dispersion curves on electric boundary conditions is obvious from the behaviour of the curves for the evanescent waves.  Fig. 6 -9. Again as for the case 1 m = the substantial difference in the dispersion curves behaviour is explained by different types of the electric boundary conditions as well as by the difference in the electro-mechanic coupling coefficients. Dispersion curves of non-axisymmetric waves with the circumferential wavenumber 3 m = in the cylinder made from PZT-4 and PZT-7A with the open-and close-circuit lateral surface are depicted in Fig. 10-13. The dispersion curves of higher circumferential wavenumbers (m = 2, 3) are sensitive to the nature of the electric boundary condition as well as to the measure of the electro-mechanical cross-coupling for both propagating and evanescent waves. These dispersions curves obtained from the exact solution of the problem could be used as references data for developing of reliable finite elements for approximate solution of the problems of wave propagation in piezoelectric structures.

Analysis of Axisymmetric and Non-Axisymmetric Wave Propagation in a Homogeneous
For electric boundary condition 1 0  For electric boundary condition 0 In the axisymmetric case we assume 0 m = in the above mentioned expressions and remove third row and fourth column in determinant (37).

Conclusion
The characteristic equation of non-axisymmetric propagating and evanescent waves of a piezoelectric cylinder of transversely isotropic material was developed. The results were numerically illustrated for sample PZT-4 and PZT-7A cylinders for the first three circumferential wavenumbers (m = 1, 2, 3) and for the axisymmetric case. It was shown that the dispersion curves are sensitive both to the electric boundary conditions and to the measure of electro-mechanical coupling. This effect was revealed more strongly in the higher order modes.

Publisher InTech
Published online 01, January, 2010 Published in print edition January, 2010

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In the recent decades, there has been a growing interest in micro-and nanotechnology. The advances in nanotechnology give rise to new applications and new types of materials with unique electromagnetic and mechanical properties. This book is devoted to the modern methods in electrodynamics and acoustics, which have been developed to describe wave propagation in these modern materials and nanodevices. The book consists of original works of leading scientists in the field of wave propagation who produced new theoretical and experimental methods in the research field and obtained new and important results. The first part of the book consists of chapters with general mathematical methods and approaches to the problem of wave propagation. A special attention is attracted to the advanced numerical methods fruitfully applied in the field of wave propagation. The second part of the book is devoted to the problems of wave propagation in newly developed metamaterials, micro-and nanostructures and porous media. In this part the interested reader will find important and fundamental results on electromagnetic wave propagation in media with negative refraction index and electromagnetic imaging in devices based on the materials. The third part of the book is devoted to the problems of wave propagation in elastic and piezoelectric media. In the fourth part, the works on the problems of wave propagation in plasma are collected. The fifth, sixth and seventh parts are devoted to the problems of wave propagation in media with chemical reactions, in nonlinear and disperse media, respectively.
And finally, in the eighth part of the book some experimental methods in wave propagations are considered. It is necessary to emphasize that this book is not a textbook. It is important that the results combined in it are taken "from the desks of researchers". Therefore, I am sure that in this book the interested and actively working readers (scientists, engineers and students) will find many interesting results and new ideas.

How to reference
In order to correctly reference this scholarly work, feel free to copy and paste the following: Michael Yu. Shatalov (2010). Analysis of Axisymmetric and Non-Axisymmetric Wave Propagation in a Homogeneous Piezoelectric Solid Circular Cylinder of Transversely Isotropic Material, Wave Propagation in Materials for Modern Applications, Andrey Petrin (Ed.), ISBN: 978-953-7619-65-7, InTech, Available from: http://www.intechopen.com/books/wave-propagation-in-materials-for-modern-applications/analysis-ofaxisymmetric-and-non-axisymmetric-wave-propagation-in-a-homogeneous-piezoelectric-solid-© 2010 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike-3.0 License, which permits use, distribution and reproduction for non-commercial purposes, provided the original is properly cited and derivative works building on this content are distributed under the same license.