Three-dimensional free vibration analysis of triclinic piezoelectric hollow cylinder

Accepted Manuscript Three-dimensional free vibration analysis of triclinic piezoelectric hollow cylinder V. Rabbani, A. Bahari, M. Hodaei, P. Maghoul, N. Wu PII:

S1359-8368(18)32614-3

DOI:

10.1016/j.compositesb.2018.09.033

Reference:

JCOMB 6003

To appear in:

Composites Part B

Received Date: 14 August 2018 Revised Date:

13 September 2018

Accepted Date: 15 September 2018

Please cite this article as: Rabbani V, Bahari A, Hodaei M, Maghoul P, Wu N, Three-dimensional free vibration analysis of triclinic piezoelectric hollow cylinder, Composites Part B (2018), doi: https:// doi.org/10.1016/j.compositesb.2018.09.033. 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.

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Three-Dimensional Free Vibration Analysis of Triclinic Piezoelectric Hollow Cylinder V. Rabbania,∗, A. Baharia , M. Hodaeia,b , P. Maghoulc , N. Wua,∗

Sound & Vib. Lab., Dep. of Mech. Eng., University of Manitoba, Winnipeg, Canada b Dep. of Biomedical Eng., University of Manitoba, Winnipeg, Canada c Dep. of Civil Eng., University of Manitoba, Winnipeg, Canada

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Triclinic materials are categorized as anisotropic elastic materials with no existence of a symmetry plane. To characterize such materials, 21 elastic constants are required. The previous studies have not investigated triclinic materials due to a high number of material constants considered in the modeling. This paper presents a closed-form 3D piezoelectric model to investigate the free vibration of an arbitrary thick triclinic piezoelectric hollow cylinder. The piezoelectric cylinder is assumed to be infinitely long and short circuit boundary conditions are applied at the inner and outer surfaces of the shell. The natural frequencies of the cylinder are calculated using the transfer matrix approach along with the state space method. The effects of different anisotropic piezoelectric properties including orthotropic, monoclinic, and triclinic materials on the dispersion curve of natural frequencies are studied. The numerical results show that if the value of the axial wave number, the circumferential wave number, or natural frequency increase the resonant frequency of triclinic material deviates from other anisotropic materials such as orthotropic. Finally the validity of the proposed model is confirmed by comparing with simplified cases studied in the literature.

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Keywords: Exact Theory of Linear Piezoelectricity, Fully Anisotropic, Triclinic, Monoclinic, Ferquency Reponse

1. Introduction The exceptional electro-mechanical coupling properties of piezoelectric materials plays a preeminent role in the advancement of various electro∗

Corresponding author

Preprint submitted to Journal Name

September 21, 2018

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mechanical equipment (Anderson and Hagood [2], Arnau et al. [3], Brown [10], DeReggi et al. [15], Jalili [27], Park and Shrout [35], Shodja and Ghazisaeidi [40]). These applications can be classified as

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• Nondestructive testing, which utilizes piezoelectric transducers in an intelligent monitoring of infrastructure such as pressure vessels, natural gas pipelines, railways, roads and bridges (Bickerstaff et al. [6], Giurgiutiu and Cuc [18], Krautkr¨amer and Krautkr¨amer [30], Rose [37]). Also, oxyborate piezoelectric crystals are used in condition monitoring in high temperatures such as turbines and furnaces (Yu et al. [49, 50, 51], Zhang et al. [52, 53]);

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• Underwater acoustics, which applies piezoelectric hydrophone in modern towed sonar array for underwater surveillance (Brown et al. [9]); • Diagnostic ultrasound, which uses small piezoelectric transducer placed within the cylinder frequently used for in-vivo ultrasound tomography for early detection of malignant lesions such as in breast and prostate tissues (Opielinski et al. [33], Opieli´ nski et al. [34], Yang et al. [47]);

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• Sensors and actuators, which use piezoelectrics for active vibration and noise control of smart structures (Hasheminejad and Alaei-Varnosfaderani [20, 21], Hasheminejad and Keshavarzpour [22], Hasheminejad et al. [25], Tzou and Gadre [41], Wu et al. [45, 46], Yang et al. [48]).

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Among all different shapes of engineering structures, the hollow cylindrical shape has attracted many attentions and been recommended where acoustic directivity, simplicity, and size constraints are the primary design concerns (Liu et al. [31], Qin et al. [36], Wang et al. [43], Zhou et al. [54]). Thus, numerous studies have been carried out on the dynamic behavior of piezoelectric cylinder (Bisheh and Wu [7], Chen et al. [12], Hasheminejad et al. [23], Hasheminejad and Rajabi [24], Hasheminejad et al. [25]). Numerous studies on the free vibration analysis of a piezoelectric cylindrical shell with some degrees of anisotropy have been done by using fully numerical methods such as FEM as well as analytical approaches. Current review, however, is focused merely on the analytical and semi-analytical approaches. The analytical solutions are based on two different approaches in describing the elastic deformation of a shell: one is based on a thin shell theory whereas the other uses the exact theory of linear elasticity. A brief review on the most related studies on the application of the thin shell theory to study the free vibration of piezoelectric structures is given next. 2

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Numerous investigations on the free vibration of different piezoelectric structures have been conducted by utilizing different thin shell theories (Bisheh and Wu [8]). Haskins and Walsh [26] utilized the traditional shell theory to investigate the free vibration of the transversely isotropic hollow piezoelectric cylindrical shell. Their numerical results obtained for the radially polarized piezoelectric cylindrical shell displayed a good agreement with experimental data in the case of a very small thickness. Drumheller and Kalnins [17] employed shell theories to develop an analytical solution for free vibration of piezoelectric cylindrical shells. Their proposed model can also satisfy both mechanical and electrical governing equations within the border of shell theories. Babaev et al. [4] utilized thin shell theory to investigate the dynamic response of radially polarized cylindrical piezoelectric shell filled and submerged with the acoustic fluid, excited by electrical signals. Babaev and Savin [5] used Kirchhoff-Love theory to study the transient vibro-acoustic response of two infinitely long coaxial piezoelectric cylinders excited by a time-dependent electrical signal. Tzou and Zhong [42] proposed a generic linear shell theory to investigate the dynamic response of thin or moderately thick piezoelectric cylinders. The suggested piezoelectric shell theory is so general that can be employed for other engineering structures, such as the sphere, plate, panel, etc. Sheng and Wang [38] used the Hamiltons principle and Maxwell’s equation along with the first order shear deformation theory to study the thermo-elastic vibration of moderately thick functionally graded piezoelectric shell under thermal loading and electrical voltage. Studies based on the exact theory of linear elasticity is reviewed next. Earlier research by the exact theory of linear piezoelasticity has attracted more attention due to its capability for obtaining accurate results while the thickness of cylinder can vary from thin to extremely thick along radius. Kapuria et al. [28] presented an analytical approach to calculate the harmonic response of simply supported orthotropic piezoelectric cylinder subjected to non-axisymmetric electromechanical loads. Chen and Shen [11] used power series expansion method to study both direct and inverse piezoelectric effect on the free vibration of a finite length orthotropic piezoelectric circular cylindrical shell. Chen et al. [12] developed a closed-form solution to study the free vibration of an arbitrarily thick functionally graded orthotropic piezoelectric hollow cylinder filled with ideal fluid. Shlyakhin [39] utilized the vector eigenfunction expansion method to find a closed-form solution for the dynamic vibration of a radially polarized orthotropic piezoelectric cylinder. More recently, Wang et al. [44] both theoretically and experimentally investigated the free vibration of a finite quartz cylinder using the 3

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Rayleigh-Ritz method and Chebyshev polynomials. The knowledge gap and subsequent proposed research is discussed next. There are numerous studies on the vibration of a cylindrical shell with some degrees of anisotropy where most of them are on transversely isotropic (Haskins and Walsh [26]) and orthotropic (Chen et al. [12], Kapuria et al. [28], Shlyakhin [39]). The need for improvement in a piezoelectric element’s sensitivity, however, requires development of materials with stronger anisotropy such as lead Lithium Tantalate, Lithium Niobate and Oxyborate crystals of Y Ca4 O(BO3 )3 (YCOB) as indicated in previous studies, for example Han and Yan [19], Kar-Gupta and Venkatesh [29], Luo et al. [32], Yu et al. [49]. The aforementioned comprehensive literature review suggests that a dispersion analysis of a fully anisotropic hollow cylindrical peizoelectric shell is yet to be addressed. Consequently, the specific novelties and highlights of this research are • monoclinic material with one symmetry plane;

• fully anisotropic or triclinic materials are the most general form of anisotropic materials without any axis of material symmetry;

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• using exact theory of elasticity rather than approximate shell theories for all ranges of thicknesses and frequencies.

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The solution of such problems includes complexity because of the system of coupled partial differential equation along with satisfying the boundary conditions on the internal and external surface of cylinder. These complexity can increase when the cylinder is made of an anisotropic material. The main aim of this study is to develop an analytical model to study, discuss and explain the free vibration of a triclinic infinite cylinder in order to fill the above-mentioned gap. The suggested solution is very important in view of the classical structural problem. The proposed method can also offer an analytical foundation for exploiting the potential ability of triclinic piezoelectric material in the intelligent cylindrical structures. It can also be used as a benchmark for comparison to other results achieved by numerical or semi-analytical methods. The specific organization of this paper is outlined as follows. The mathematical modelling section is developed based on the theory of linear piezoelasticity. This includes the constitutive relations of exact three dimensional (3D) elastic medium, kinematic assumptions, conservation laws, traction boundaries and solution assembly. Next, the traction boundary conditions and final solution are obtained. Then the validity of proposed mode was 4

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checked against other literature’s results. Next, the effect of anisotropy on the wave dispersion curves of piezoelectric cylinder are discussed in the case studies section. Finally, the conclusion remarks are listed. 2. Mathematical Modelling

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A detailed description of the problem is given first. It is important to note that the current approach is applicable to any fully anisotropic material and different piezoelectric polarization cases (i.e. radially, axially and circumstantially polarized). 2.1. Problem Description

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An infinitely long piezoelectric cylindrical shell is modeled by a fully triclinic anisotropic assumption. The cylindrical shell is made of a triclinic piezoelectric material with internal and external radius of a and b, respectively, as shown in Figure 1. Constitutive relations are given next.

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Figure 1: Problem Configuration.

2.1.1. Constitutive Relations The general, linear, anisotropic constitutive relations for piezoelasticity can be written as (Chopra and Sirohi [14]) σij = Cijkl kl − eijk Ek ,

Di = eikl kl + κij Ej ,

(1)

where eijk , σij , and kl , are the third order piezoelectric tensors, second order Green-Cauchy stress and strain tensor which have 18, 6 and 6 independent parameters, respectively. Cijkl and κij are the fourth order elasticity and 5

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Ei = −φ,i ,

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second order dielectric permittivity tensors measured under the conditions of zero electric and strain fields which have up to 21 and 6 independent parameters, respectively. Thus a comprehensive description of a triclinic piezoelectric material involves the recognition of all 45 independent constants. The expanded matrix form of Equation 1 are given in Appendix A. Furthermore, Ei and Di represent the electrical field and the electrical displacement vectors, respectively. The electrical field Ei can be defined by using an electrical potential, φ, such that (2)

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→ − where ,i ≡ ∇ is the forward gradient operator in cylindrical coordinates which is given in Appendix B. The kinematic assumptions will be discussed next. 2.1.2. Kinematic Assumptions The ij is the Lagrangian finite strain tensor in which its linearized form for infinitesimal deformations, ui , turns into 1 ij = (ui,j + uj,i ), 2

(3)

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→ − where ui indicates an infinitesimal deformation. Moreover ,i ≡ ∇ and ← − ,j ≡ ∇ are the forward and backward Nabla operators. The expanded matrix form of Equation 3 is given in Appendix C. The Conservation Laws will be disscussed next.

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2.1.3. Conservation Laws Conservation of linear momentum for a differential element can be expressed as σij,j + fi − ρu¨i = 0,

(4)

where fi , ρ and u¨i are a body force, unperturbed material density and the Lagrangian particle acceleration, respectively. The first term, divergence of stress tensor, is determined by inner product of the second order stress tensor with the cylindrical gradient operator which is given in Appendix B. The expanded form of Equation 4 can be found in Equation D.1. The electrostatic charge equilibrium of a piezoelectric material can be

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expressed as (5)

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Di,i = Qf ,

where Qf is the free charge density. Hence the equilibrium conditions for a piezoelectric material are satisfied by implementation of both of Equation 4 and Equation 5. The expanded form of Equation 5 is given in Equation D.2. The traction boundaries are given next.

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ti = σij nj ,

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2.1.4. Traction Boundaries The mechanical boundary conditions require the continuities of traction forces, ti , and displacement ui vectors. The traction force can be defined as (6)

where ti and ni are the Cauchy stress vector at a traction surface and unitlength direction vector, respectively. The mixed mechanical and electrical boundary conditions, under the assumption of perfect bounding, require the continuities of displacement ui , traction force ti , electric potential φi and the electrical displacement Di . As the centre of the cylindrical coordinate is located at the center-line of the shell, en ≡ er , hence,

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t = σrr er + σrθ eθ + σrz ez ,

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where en is the unit base vector. The solution assembly will be discussed next. 2.2. Solution Assembly

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The field’s variables can be expanded in terms of trigonometric functions in the circumferential θ and axial z directions in a cylindrical coordinate, such that V=

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in which n, e, ζ and ω are the circumferential wave number, the exponential function, the axial wave number and the circular frequency, √ respectively. Furthermore V ≡ V(r, θ, z, ω) is spatial state vectors and i = −1. The

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modal components of the aforementioned sate vector, Vn (r, ω), are

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 n  ur (r, ω)        unθ (r, ω)        n   u (r, ω)   z  n    σrr (r, ω) Vn (r, ω) = n (r, ω) . σrθ       n    σrz (r, ω)    n (r, ω)     D    r  φn (r, ω)

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(9)

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After performing a tedious task of substituting modal expansions given in Equation 9 into Equation 1, Equation 2, Equation 3 and Equation 4 along with the application of orthogonality properties of trigonometric functions in the absence of a body force, fi , and electrical charge density, Qf , a modal state space of the system emerges can be written as dVn (r, ω) = Ξn (r, ω)Vn (r, ω), dr

n = −∞, ..., −1, 0, 1, ..., +∞,

(10)

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where Ξn (r, ω) is a 8 × 8 modal coefficient matrix whose elements are given in Appendix E. A general analytical solution of Equation 10 is not straightforward when Ξ is position dependent. Consequently an approximate laminated model is adopted as Ding and Chen [16]. Subsequently the piezoelectric thickness is assumed to be composed of npz perfectly bonded sub-layers with equal ˆ = (b − a)/npz . The elements of matrix Ξn (r, ω) is assumed thicknesses of h to be constant and equal to the values at the mid-surfaces r¯k = (rk +rk−1 )/2 ˆ  1. Accordingly each because of very small thickness of each sub-layer, h layer’s solution to Equation 10 is

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Vn (r, ω) = e((r−rk−1 )Ξn (¯rk ,ω)) Vn (rk−1 , ω), ˆ , rk = a + k h, ˆ rk−1 = a + (k − 1)h

rk−1 ≤ r ≤ rk ,

(11)

k = 1, 2, ..., npz .

Evaluating the latter equations for the k th layer leads to ˆ Vn (rk , ω) = e(Ξn (¯rk ,ω)h) Vn (rk−1 , ω).

(12)

Global transfer matrix is obtained, next, by enforcing the continuity of Vn between all interfaces. Then the modal components of the state 8

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Vn (b, ω) = Θn (ω)Vn (a, ω), npz

Θn (ω) =

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variables, Vn , at the outer and inner radii are related through the global transfer matrices, Θn , such that

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for the piezoelectric layer where Θn (ω) is 8 × 8 ultimate transfer matrix, respectively. The traction boundary conditions and final solution will be discussed next. 3. Traction Boundary Conditions and Final Solution

The tangential and normal stresses at the internal and external surfaces of the piezoelectric cylinder can be easily expressed as Chen and Ding [13]:     n σrr (r, ω) 0 n (r, ω) r = a, b. (15) = 0 , tn ≡ σrθ     n 0 σrz (r, ω)

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The short-circuit electrical boundary condition at inner and outer surfaces of the piezoelectric cylinder can be written as φn (r, ω)|r=a = φn (r, ω)|r=b = 0.

(16)

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Next, by substitution of boundary conditions (Equation 15 and Equation 16) into the global transfer function (Equation 13), it is obtained,   ur (b, ω)        uθ (b, ω)          u (b, ω)   z     0 = Θn (ω) 0           0         D (b, ω) r     0

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  ur (a, ω)        uθ (a, ω)          u (a, ω)   z     0 . 0           0         D (a, ω) r     0

(17)

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By rearranging the system of Equation 17, the homogeneous linear system of algebraic equations can be written as Ax = 08×1 ,

(18)

in which |A| is the determinant of matrix A.

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where, 08×1 is a zero vector. The expanded form of Equation 18 can be found in Appendix F. By solving the eigenvalue problem of Equation 18, a non-trivial frequency domain equation of triclinic piezoelectric cylinder is obtained as A = 0, (19)

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Non-dimensional axial wave number,

Figure 2: Non-dimensional wave phase velocity Υ versus the non-dimensional axial wave number ξ for (b-a)/a=0.05.

4. Validations

Due to the wide-range of different configurations and parameters offered in the problem formulation, while keeping in mind the computational cost and restrictions, we should focus on some logically selected model configurations. In order to show that the proposed model can handle very thick 10

ij (10−11 F/m)

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T riclinic 2.385 0.768 0.633 0.06 0.028 0.005 2.412 0.637 −0.027 −0.010 −0.002 2.177 −0.027 −0.019 −0.003 0.860 0.057 0.055 0.843 0.053 0.764 −0.614 0.117 0.122 −0.053 −0.727 −1.252 −0.1 −0.906 0.044 −1.433 −0.094 1.908 −0.811 −1.202 2.043 −1.274 2.046 0.045 31.753 −0.089 1.780 −2.123 −2.123 43.846 6075

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Y Ca4 O(BO3 )3 1.551 0.142 0.422 0 0 0 1.451 −0.051 0 −3.5 0 1.244 0 0.016 0 0.29 0 0.007 0.478 0 61.4 0.14 0 −0.47 0 −0.07 −2.6 0 −1.9 0 0.13 0 0.49 −0.11 −0.36 0.15 −2.6 −0.23 0 8.540 0 0.840 10.443 0 8.451 3310

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LiT aO3 2.33 0.80 0.47 0.11 0 0 2.75 0.80 0 0 0 2.330 −0.11 0 0 0.94 0 0 0.93 0.11 0.94 0 0 0 0 −1.6 −2.6 0 −1.9 0 0 0 0 −1.6 0 1.6 −2.6 0 0 36.3 0 0 38.1 0 36.3 7450

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LiBnO3 2.45 0.75 0.75 0 0 0 2.03 0.53 0 0.09 0 2.03 0 0.09 0 0.75 0 0.09 0.6 0 0.6 1.3 0.2 0.2 0 0 0 0 0 0 −2.5 0 3.7 0 −2.5 2.5 0 3.7 0 25.7 0 0 38.9 0 38.9 4700

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P ZT 4 1.39 0.78 0.74 0 0 0 1.39 0.74 0 0 0 1.15 0 0 0 0.256 0 0 0.256 0 0.305 0 0 0 0 12.7 0 0 0 0 12.7 0 0 −5.2 −5.2 15.1 0 0 0 650 0 0 650 0 560 7500

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C11 C12 C13 C14 C15 C16 C22 C23 C24 C25 C26 C33 C34 C35 C36 C44 C45 C46 C55 C56 C66 e11 e12 e13 e14 e15 e16 e21 e22 e23 e24 e25 e26 e31 e32 e33 e34 e35 e36 κ11 κ12 κ13 κ22 κ23 κ33 ρ

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Table 1: Mechanical and electrical properties of the constituent materials.

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geometries, the inner and outer radius of piezo-composite cylinder are considered as a = 1 m and b = 1.5 m. A general MATLABr parallel code was established in order to determine the final global transfer matrix Θn as well as attaining matrix determinant (Equation 19). The calculation was performed on the cluster of Intelr Xeonr Processor E5-2630 v4 desktop computer (15M Cache, 2.30 GHz, 7.20 GT/s) which benefits from parallel core technology for multi-threaded applications. By utilizing Matlab Parallel Computing Toolbox we optimized our code to be able to exploit the full processing power of multi-core desktops by executing applications on 40 workers (MATLAB computational engines) that run locally.

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Figure 3: Non-dimensional wave phase velocity Υ versus the non-dimensional axial wave number ξ for (b-a)/a=0.5.

Before addressing the main result, the overall accuracy of the proposed solution shall be studied. In the current study on an infinite length cylinder without certain boundary, it is not easy to use numerical method such as FEM to directly simulate and describe the characteristics of wave behaviors in the infinite structures with different parameters of material properties and structural sizes. Thereby, other literature have been used to check the 12

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PZT4 LiBnO3

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YCa4O(BO3)3

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Figure 4: Non-dimensional wave phase velocity Υ versus the non-dimensional axial wave number for selected piezoelectric materials material n=0 and (b − a)/a = 0.5.

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validity of the results. First, the cylinder is assumed to be fabricated from non-graded PZT4. The elastic constants Cij , piezoelectric constants eij , dielectric permittivity constants κij and density of different piezoelectric materials are provided in Table 1. In order to satisfy the convergence of final global transfer matrix of the cylindrical piezoelectric cylinder, we used thirty (npz = 30) sub-layers to calculate the final global qtransfer matrix. The lowest non-dimensional natural frequency, Ψ = ωa Cρ44 , was determined. Then, the non-dimensional wave phase velocity Υ =

Ψ ξ

versus the non-dimensional axial wave num-

ζ a

ber ξ = were depicted in Figure 2 and Figure 3 for a relatively thin (b−a) = 0.05 and thick (b−a) = 0.5 hollow cylinder, respectively. The short a a circuit boundary condition is applied at both cylindrical surfaces. The outputs as shown in the Figure 2 and Figure 3 display a very good agreement with Chen et al. [12]. It is clear that as the value of non-dimensional axial wave number in13

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Figure 5: Non-dimensional wave phase velocity Υ versus the non-dimensional axial wave number for n=1 and (b − a)/a = 0.5.

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creases, the curves for all different flexural modes gradually become invariant as well as approaching to a constant value. Furthermore, the behavior of dispersion curves for the same flexural mode are different in thick and thin piezoelectric cylinder. For instance, for the mode number n = 2, the value of non-dimensional wave phase velocity, υ first decreases and then increases and then decreases to constant value when (b − a)/a = 0.05, while it first decreases and then increases and approaches to a constant value when (b − a)/a = 0.5. Consequently, the results for a thin cylindrical shell do not follow the same pattern of those of thick cylindrical shell. The case studies will be discussed next. 5. Case Studies The configuration and material selection is based on the thick case which have been addressed in section 4. Figure 4 and Figure 5 show the disper14

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sion curves of the non-dimensional wave phase velocity Υ versus the nondimensional axial wave number ξ for the five different piezoelectric materials, as listed in Table 1. Although all the plots finally become almost invariant at high wave numbers larger than 15, it is obvious that there is a significant difference between the dispersion curves of PZT cylinder with the other materials. The PZT4 material is considered orthotropic with 9 independent material constants, while the other materials have more than 9 independent constants. Since the stiffness of PZT4 material is lower than the other materils such as LiBnO3 , LiT aO3 , and Y Ca4 O(BO33 ), it is expected that the Non-dimensional wave phase velocity for the PZT4 is significantly lower than the other materials. Furthermore, the dispersion curve of PZT cylinder becomes rapidly invariant by the increase in the axial wave number, while the dispersion curves for the other anisotropic materials become gradually constant with relatively higher values of non-dimentional wave phase velocities. This is due to the fact that the PZT material is classified as the orthotropic piezoelectric materials, however the other curves belong to the monolithic or triclinic piezoelectric materials which show a different pattern compare to the conventional orthotropic PZT material. This observation shows that the previous proposed theoretical methods for orthotropic piezoelectric material can not be simply extended to monolithic or triclinic piezoelectric materials which need to take into consideration a higher number of non-zero material constants. The non-dimensional wave phase velocities of triclinic material are found to have a significant difference with other anisotropic materials as well. Figure 6 shows the dispersion curves of the lowest natural frequencies ω versus the circumferential wave number n for the triclinic and the orthotropic piezoelectric materials for selected axial wave numbers ξ. The triclinic material used in Table 1 is made up one layer of LiN bO3 with the crystalline Z axis along z and the crystalline Y axis along r and the second layer is made up LiT aO3 with crystalline X axis along z and the crystalline Y along r, in order to show the triclinic effect on the wave behavior (see Akcakaya and Farnell [1]). In order to make the results of triclinic and orthotropic materials comparable, the orthotropic material used in Figure 6, Figure 7 and Figure 8 is made up the same triclinic material by putting all the nonzero constants to zero in the way that it changes from a triclinic material to an orthotropic piezoelectric, for example PZT4. Regarding Figure 6, the circumferential wave number has a significant effect on the natural frequencies of both triclinic and orthotropic materials. It is axiomatic that there is more sever difference between the natural frequencies of the two 15

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0

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30

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50

60

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Circumferential Wave Number, n

Figure 6: First lowest natural frequencies ω versus the circumferential wave number n for selected axial wave number ξ and (b − a)/a = 0.5.

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materials in higher circumferential wave number, regardless of the value of axial wave number. The difference in the natural frequencies of the two materials depends on the value of the axial wave number. As it is clearly seen from Figure 6, as the value of the axial wave number ξ increases from 0.001 to 60, the difference between the natural frequencies of two materials become bigger. In addition, for the lower axial wave number such as ξ = 0.001 and high circumferential wave number, for example n = 60, the difference between the natural frequencies of the two materials is considerable, and the natural frequency of the triclinic material is higher than the one of orthotropic material by 580 Rad/s. Likewise, for the lowest circumferential wave number n = 0 and higher axial wave number ξ = 60 the natural frequency difference is also noteworthy, and the natural frequency of the triclinic material is higher than the one of orthotropic material by 3100 Rad/s. The difference between the natural frequencies of triclinic and orthotropic materials is due to the fact that the triclinic material has more non-zero material constants which results in the consideration of more stress terms with a given strain vector. It can also be concluded that the conventional orthotropic model of piezoelectric cylinder including lower number of material constants can only be used to capture the lowest natural frequency of totally anisotropic piezoelectric cylinder when both axial wave number and circumferential wave number are relatively low. In other word, the triclinic model proposed in this research which considers all possible constants should be used to study the triclinic piezoelectric cylinder for higher axial wave numbers or higher circumferential wave numbers because the natural frequency of orthotropic model deviates from the triclinic model, obviously. Figure 7a and Figure 7b display the dispersion curves of the natural frequency versus the non-dimensional axial wave number ξ for selected lowest natural frequencies while the circumferential wave number is equal to zero. Results for the twelve selected natural frequencies show that the difference between the natural frequencies of the two materials is very small for the first lowest natural frequency. However, for higher natural frequencies, the difference between the natural frequencies of two materials become gradually bigger and the two plots become completely different in higher natural frequency, such as shown in the figure, for the 55th lowest natural frequency, and the natural frequency of triclinic material is higher than the one of orthotropic material by 4300 Rad/s. This observation indicates that the conventional orthotropic model of piezoelectric cylinder with a lower number of non-zero material constants can only be used to model the anisotropic piezoelectric materials when the value of the natural frequency is relatively 17

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3.5

5.5

3

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2.5

4.5 2 4 1.5 1

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1.65 1.64 1.63 1.62

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Non-dimensional Axial Wave Number ( ξ)

Figure 7a: Natural frequencies ω versus the non-dimensional wave number ξ for selected lowest natural frequency with n=0 and (b − a)/a = 0.5.

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2.97

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2.485

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2.465 2.93 2.46

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2.445

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Non-dimensional Axial Wave Number ( ξ)

Figure 7b: Natural frequencies ω versus the non-dimensional wave number ξ for selected lowest natural frequency with n=0 and (b − a)/a = 0.5.

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small. So, the value of the lowest natural frequency is the third important factor after the circumferential wave number and the axial wave number when the orthotropic model is utilized to predict the dynamic behavior of a triclinic piezoelectric materials. As Figure 7 shown, the value of natural frequency of orthotropic material is gradually increasing in all figures while it decreases and then increases at 10th lowest natural frequency. Figure 8 shows the cut-off frequency of first eight natural frequencies, ω, versus the non-dimensional axial wave number, ξ, for the circumferential wave number n = 0. It can be seen that the difference between the dispersion curves of othotropic and triclinic materials increase when the fundamental frequencies order change to higher values. For instance, while the two plots are exactly matched in the first and second order frequencies, a significant difference can be observed in the 7th and 8th order frequencies.As Figure 8 shown, In lower frequencies such as 1st to 4th the triclinic and orthotropic material have the same behaviour. As a result, the simple models of ortothropic materials with three plane of symmetry and lower number of material constant can be used instead of the complicated triclinic material model. 10

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3 2

8th 7th

6th

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4th

2nd

3th

1st

1 0

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10

Non-dimentional axial Wave Number, ( ) Figure 8: Cut-off frequency of the lowest natural frequencies ω versus the non-dimensional axial wave number ξ for n = 0 and (b − a)/a = 0.5.

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6. Conclusions

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(1) The state space method, transfer matrix approach along with appropriate state vector expansion are utilized to study the free vibration of an arbitrary thick, infinitely long triclinic hollow piezoelectric cylinder. The free vibration of the most general form of an anisotropic piezoelectric material with all possible material constants is analytically modeled and studied to cover a broad range of anisotropic piezoelectric materials. (2) Numerical results demonstrate that the orthotropic model can only be used to model triclinic piezoelectric materials when the circumferential wave number, axial wave number, and the value of the lowest natural frequency are all relatively small. So, if one of these three important parameters increases, the results of the orthotropic piezoelectric model can not be extended to investigate the behavior of the triclinic piezoelectric materials. (3) Since no assumption was considered for stress and displacement fields, the output results can be used as a benchmark to investigate the accuracy of different shell models, semi-numerical and numerical solutions. (4) As part of the future works, the proposed model can also be easily used to study the free vibration of functionally graded anisotropic material. Furthermore, due to considering all the possible material constants for the general anisotropic piezoelectric, the model are able to investigate the natural frequencies of monolithic piezoelectric material. The model can be extended to studies of multi-layers anisotropic piezoelectric structures. In addition, it can be used to investigate the effect of different polarization directions on the natural frequencies of general anisotropic piezoelectric materials.

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Acknowledgment

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This research was undertaken, in part, thanks to funding support from the University of Manitoba, Research Manitoba, and Natural Sciences and Engineering Research Council of Canada (NSERC).

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Appendix A

Anisotropic Piezoelectric Constitutive Model in an ex-

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The constitutive equation for a triclinic piezoelectric material panded matrix form can be written as       σ C C C C C C rr  e11 e21 rr 11 12 13 14 15 16                   σθθ   . C22 C23 C24 C25 C26     θθ   e12 e22      zz  e13 e23 σzz . . C C C C 33 34 35 36  = −  . e14 e24 σθz  . . C44 C45 C46  θz                     e15 e25     σ . . . . C C  rz 55 56 rz         σrθ . . . . . C66 rθ e16 e26

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 e31   e32   Er   e33  Eθ e34   Ez  e35  e36 (A.1)

Differential Operator

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Appendix B

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   rr                θθ  D e e e e e e κ11 κ12 κ13 Er   r    11 12 13 14 15 16  D =  e21 e22 e23 e24 e25 e26  zz + . κ22 κ23  Eθ . θz   θ      Dz e31 e32 e33 e34 e35 e36 . . κ33 Ez          rz   rθ (A.2) Equation A.1 and Equation A.2 are corresponding to those of Equation 1.

The cylindrical gradient operator, ∇, is

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∇=

Appendix C

1 ∂ ∂ ∂ er + eθ + ez . ∂r r ∂θ ∂z

(B.1)

Kinematic Relations

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The Cauchy-Green infinitesimal deformation tensor in an expanded matrix form becomes   
 uθ ∂uθ ∂ur ∂ur 1 1 ∂ur 1 ∂uz − + + r ∂r 2 ∂r ∂z     ∂r 2 r ∂θ   rr rθ rz     . θθ θz  =  . ur ∂uz  . (C.1) 1 ∂uθ 1 ∂uθ  r ∂θ + r 2 ∂z + r∂θ    . . zz   .

.

Equation C.1 corresponds to the one of Equation 3. 22

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Appendix D

Conservation Laws

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The expanded matrix form of conservation of linear momentum in the absence of body force is given as

(D.1a)

1 ∂ 1 ∂σθθ ∂σθz σrθ (rσrθ ) + + + = ρ¨ uθ , r ∂r r ∂θ ∂z r

(D.1b)

1 ∂σθz ∂σzz 1 ∂ (rσrz ) + + = ρu¨z , r ∂r r ∂θ ∂z

(D.1c)

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1 ∂ 1 ∂σrθ ∂σrz σθθ (rσrr ) + + − = ρu¨r , r ∂r r ∂θ ∂z r

where a double over-dot represents the second order differentiation with respect to time. Equation D.1 corresponds to the one of Equation 4. The electrostatic charge equilibrium of a piezoelectric material in the absence of free charge density is written as 1 ∂ (rDr ) 1 ∂Dθ ∂Dz + + = 0. r ∂r r ∂θ ∂z

(D.2)

Modal Coefficient Matrices

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Appendix E

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Equation D.2 corresponds to the one of Equation 5.

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The modal coefficient elements of piezoelectric medium, Ξ, can be written as   Ξ(1, 1) · · · Ξ(1, 8)  ..  , .. Ξ8×8 =  ... (E.1) . .  Ξ(8, 1) · · · Ξ(8, 8)

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where

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Ξ(1, 1) = −C25 C66 e11 e51 − C15 C66 e21 e51 − C16 C26 e251 + C12 C66 e251 + C16 C25 e51 2 e61 +C15 C26 e51 e61 −C15 C25 e261 −C15 C25 C66 11 −C56 (C12 11 + e11 e21 )+C55 (e61 (−C16

e21 +C12 e61 ) + C66 (C12 11 + e11 e21 ) − C26 (C16 11 + e11 e61 ))+C56 ((C25 e11 + C15 e21

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(C15 11 + e11 e51 ) ,

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− 2C12 e51 ) e61 + C26 (C15 11 + e11 e51 ) + C16 (C25 11 + e21 e51 ))) / (r (2C15 C66 e11 e51
2 2 2 2 2 2 C56 C11 11 + e211 +C15 e61 −2C16 C15 e51 e61 +C16 e51 −C11 C66 e251 +C66 C15 11 −C55 C66
2 2 2 C11 11 + e11 −2C16 e11 e61 +C11 e61 +C16 (−11 )−2C56 (e61 (C15 e11 − C11 e51 ) +C16

.. .

Ξ(1, 8) = i C16 e51 e52 e61 n + C16 e51 e53 e61 ζr + C55 e12 e261 n + C55 e13 e261 ζr− C15 e52 e261 n−C15 e53 e261 ζr−C16 e251 e62 n−C55 e11 e61 e62 n+C15 e51 e61 e62 n−C16 e251 e63 ζr−C55 e11 e61 e63 ζr+C15 e51 e61 e63 ζr−C16 C55 e62 n11 −C16 C55 e63 ζr11 +C16 C55 e61

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n12 +C16 C55 e61 ζr13 +C66 (e11 e51 e52 (−n) − e11 e51 e53 ζr − C15 e52 n11 − C15 e53 ζ

r11 +e12 n C55 11 + e251 + −C55 e11 n12 + C15 e51 n12 + e13 ζr C55 11 + e251 +

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ζr13 (C15 e51 − C55 e11 )+C56 (−2e12 e51 e61 n − 2e13 e51 e61 ζr + e11 e52 e61 n + e11 e53 e61 ζr+e11 e51 e62 n+e11 e51 e63 ζr+C16 e52 n11 +C16 e53 ζr11 +C15 e62 n11 +C15 e63

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2 2 ζr11 −C16 e51 n12 −C15 e61 n12 − r 2C15 C66 e11 e51 + C16 e51 − C11 C66 e251 − 2C15

2 2 2 2 C16 e51 e61 +C15 e61 +C15 C66 11 +C56 C11 11 + e211 −C55 C66 C11 11 + e211 − 2

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2 C16 e11 e61 +C11 e261 +C16 (−11 )− 2C56 (C16 (C15 11 + e11 e51 ) + e61 (C15 e11 − C11

e51 )))), .. .

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2 2 Ξ(8, 1) = −C12 C56 e11 + C12 C55 C66 e11 + C11 C56 e21 − C11 C55 C66 e21 − C11 C26 C56 2 2 e51 +C11 C25 C66 e51 +C16 (C55 e21 − C25 e51 )+C11 (C26 C55 − C25 C56 ) e61 +C15 (C66 e21 −

C26 e61 +C16 (C56 (C25 e11 − 2C15 e21 + C12 e51 ) + C26 (C15 e51 − C55 e11 ) + (C15 C25 −

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C12 C55 e61 + C15 (C26 C56 e11 − C66 (C25 e11 + C12 e51 ) + C12 C56 e61 ))/ (r (2 C15 C66 e11
2 2 2 2 2 2 e51 +C16 e51 −C11 C66 e251 −2C15 C16 e51 e61 +C15 e61 +C15 C66 11 +C56 C11 11 + e211 −C55

2 C66 C11 11 + e211 − 2C16 e11 e61 + C11 e261 + C16 (−11 ) −2C56 (e61 (C15 e11 − C11 e51 + C16 (C15 11 + e11 e51 )))),

Ξ(8, 8) = −

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2 i C15 C66 e12 e51 n + C15 C66 e13 e51 ζr + C15 C66 e11 e52 n + C16 e51 e52 n

2 −C11 C66 e51 e52 n+C15 C66 e11 e53 ζr+C16 e51 e53 ζr−C11 C66 e51 e53 ζr−C15 C16 e52 e61 n− 2 2 2 C15 C16 e53 e61 ζr−C15 C16 e51 e62 n+C15 e61 e62 n−C15 C16 e51 e63 ζr+C15 e61 e63 ζr+C15 C66 2 2 n12 +C15 C66 ζr13 +C56 (C11 (n12 + ζr13 ) + e11 (e12 n + e13 ζr))+C55 (−C11 e61 (e62

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2 n+e63 ζr+C16 (e11 (e62 n + e63 ζr) + e12 e61 n + e13 e61 ζr)+.C16 (n12 + ζr13 )−C66 (

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C11 (n12 + ζr13 )+e11 (e12 n+e13 ζr)))−C56 (C15 (e11 (e62 n + e63 ζr) + e12 e61 n + e13 e61 ζr−C11 (e51 (e62 n + e63 ζr) + e52 e61 n + e53 e61 ζr)+(C15 (e11 (e62 n + e63 ζr) + e12 e61 n+e13 e61 ζr−C11 (e51 (e62 n + e63 ζr) + e52 e61 n + e53 e61 ζr)+C16 (e12 e51 n + n

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2 2 (2C15 12 + e11 e52 )+rζ (2C15 13 + e13 e51 + e11 e53 )))))/ r 2C15 C66 e11 e51 + C16 e51
2 2 2 2 2 2 −C11 C66 e51 −2C15 C16 e51 e61 +C15 e61 +C15 C66 11 +C56 C11 11 + e11 −C55 (C66 ( 2 C11 11 +e211 −2C16 e11 e61 +C11 e261 +C16 (−11 )−2C56 (e61 (C15 e11 − C11 e51 ) + C16 (C15

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11 + e11 e51 ))))).

Hint: The rest of arrays of Matrix Ξ can be found on the supplementary files.

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Appendix F

Final System of Linear Algebraic Equation

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The expanded matrix form of the transformed modal coefficients of Equation 18 can be expressed as    Θ(1, 1) Θ(1, 2) Θ(1, 3) −1 0 0 Θ(1, 7) 0  ur (a, ω)    u (a, ω)   Θ(2, 1) Θ(2, 2) Θ(2, 3) 0 −1 0 Θ(1, 7) 0     θ       Θ(3, 1) Θ(3, 2) Θ(3, 3) 0   0 −1 Θ(1, 7) 0 u (a, ω)  z       Θ(4, 1) Θ(4, 2) Θ(4, 3) 0  0 0 Θ(4, 7) 0 u (b, ω) r   = 08×1 . Θ(5, 1) Θ(5, 2) Θ(5, 3) 0 0 0 Θ(5, 7) 0  uθ (b, ω)       Θ(6, 1) Θ(6, 2) Θ(6, 3) 0  uz (b, ω)    0 0 Θ(6, 7) 0         Θ(7, 1) Θ(7, 2) Θ(7, 3) 0  Dr (a, ω)  0 0 Θ(7, 7) −1     Θ(8, 1) Θ(8, 2) Θ(8, 3) 0 0 0 Θ(8, 7) 0 Dr (b, ω) (F.1) REFERENCES

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[4] Babaev, A., But, L., Savin, V., 1990. Transient vibrations of a thinwalled cylindrical piezoelectric vibrator driven by a nonaxisymmetric electrical signal in a liquid. International Applied Mechanics 26 (12), 1167–1174.

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[21] Hasheminejad, S. M., Alaei-Varnosfaderani, M., 2014. Acoustic radiation and active control from a smart functionally graded submerged hollow cylinder. Journal of Vibration and Control 20 (14), 2202–2220. [22] Hasheminejad, S. M., Keshavarzpour, H., 2016. Robust active sound radiation control of a piezo-laminated composite circular plate of arbitrary thickness based on the exact 3d elasticity model. Journal of Low Frequency Noise, Vibration and Active Control 35 (2), 101–127.

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