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3-D waves in porous piezoelectric materials
Mechanics of Materials 80 (2015) 96–112
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3-D waves in porous piezoelectric materials Anil K. Vashishth ⇑, Vishakha Gupta Department of Mathematics, Kurukshetra University, Kurukshetra 136 119, India
a r t i c l e
i n f o
Article history: Received 1 May 2014 Received in revised form 14 August 2014 Available online 2 October 2014 Keywords: Anisotropic Phase velocity Porosity Porous piezoelectric materials Wavefront Waves
a b s t r a c t Wave propagation in porous piezoelectric materials, possessing crystal symmetries monoclinic (2, m), orthorhombic (222, 2 mm), tetragonal (4), trigonal (32), hexagonal mÞ, is studied. The Christoffel equation is derived for 3D waves in (6 mm) and cubic (43 an anisotropic porous piezoelectric medium. The four roots of the biquadratic equation give the complex wave velocities of four waves propagating in such a medium. These complex wave velocities are resolved to obtain the phase velocities and attenuation factors of waves. The algebraic implicit expressions are derived for monoclinic (2, m), orthorhom m) crystal bic (222, 2 mm), tetragonal (4), trigonal (32), hexagonal (6 mm) and cubic (43 classes. The characteristics of waves in porous piezoelectric materials are studied in terms of the velocity surfaces and attenuation surfaces. The effects of phase direction, frequency, piezoelectricity, porosity and crystal symmetry on the velocity surfaces and attenuation surfaces are investigated for these crystal classes. The effects of phase direction and crystal symmetry on the skewing angles and wave fronts of quasi waves are also studied. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Piezoelectric materials possess the important property of linear coupling between mechanical and electrical fields, which renders them useful as transducers, actuators, sensors and filters, etc. in wide range engineering applications in smart structures and devices. Wave propagation in piezoelectric media has been extensively investigated in connection with the generation and transmission of disturbances in electro-acoustic devices such as transducers and resonators. Some of the notable texts on wave propagation in piezoelectric materials are Auld (1973, 1990); Nayfeh (1995) and Royer and Dieulesaint (1996). A detailed survey of wave propagation in piezoelectric materials belonging to different crystal classes is presented in the texts (Auld, 1973, 1990; Royer and Dieulesaint, 1996). Three waves
⇑ Corresponding author. E-mail addresses: [email protected] [email protected] (V. Gupta). http://dx.doi.org/10.1016/j.mechmat.2014.09.002 0167-6636/Ó 2014 Elsevier Ltd. All rights reserved.
(A.K.
Vashishth),
viz. quasi P, quasi S1 and quasi S2 waves propagate in anisotropic piezoelectric materials. One of the essential features of the wave phenomenon in such materials is the stiffened waves which exist as a result of electromechanical coupling. The phase velocities and slowness do depend upon directions of propagation, crystal symmetry, poling directions and other features of the medium. Kyame (1949) studied the wave propagation in piezoelectric materials by taking quasi-static electric field approximation into account. The propagation of waves at large distances from a source of disturbance in an infinite piezoelectric medium of hexagonal symmetry was investigated by Rao (1978). Auld (1981) presented a short survey related to wave propagation and resonance phenomena in piezoelectric materials and relates the concepts and theory to the physical properties and crystal symmetry of the materials. Special features peculiar to wave propagation in piezoelectric materials were noted and a brief sketch of the methods used for solving piezoelectric boundary value problems was also given. It was found that an extra
A.K. Vashishth, V. Gupta / Mechanics of Materials 80 (2015) 96–112
solution appears due to piezoelectric stiffening terms when only one of the components of the wave vector is specified. Christoffel equations for electroacoustic waves in unbounded piezoelectric crystals were solved by Every (1987) for the weak and strong electromechanical coupling cases. In general, waves in piezoelectric materials are quasi waves. However, for some types of material symmetry, pure longitudinal or pure transverse wave modes exist for certain specified directions of propagations (Romeo, 1996). Daros and Antes (2000) derived the strong ellipticity conditions for piezoelectric materials of the crystal symmetry classes 6 mm and 622 using positivity conditions for quadratic, cubic and quartic polynomials. The differential equation for the wavefront shape due to a line source in cylindrically hexagonal piezoelectric solids was obtained by Daros (2000). In spite of the great advances in the development of single phase piezoelectric materials with high hydrostatic sensitivity, such materials have shortcomings such as large hydrostatic piezoelectric coefficients and flexibility, high hydrostatic sensitivity and low acoustic impedance, etc. However, due to high hydrostatic figure of merit and low acoustic impedance, porous piezoelectric materials have been of great interest and technological importance in ultrasonic applications such as hydrophones, actuators, miniature accelerometer and underwater transducers (Shrout et al., 1979; Arai et al., 1991). Use of the piezoelectric effect in porous piezoelectric ceramics offers an original method for studying the coupling between the electrical, mechanical, permeability and of course piezoelectric properties of porous systems. Different analytical models (Matsunaka et al., 1988; Banno, 1993; Gomez and Montero, 1996a) have been developed to study the effects of pore volume fraction and the connectivity on the elastic, dielectric and piezoelectric properties of porous piezoelectric materials. Experimental studies (Roncari et al., 2001; Xia et al., 2003; Boumchedda et al., 2007; Levassort et al., 2007; Zeng et al., 2007; Lee et al., 2008; Wang et al., 2008; Boumchedda et al., 2010) have been done related to characteristics, fabrication and manufacturing of porous piezoelectric materials and the influence of porosity on its properties. The study of the effects of porosity on the electromechanical properties of porous piezoelectric materials, on the basis of numerical models based on finite-element method, was done by Kar-Gupta and Venkatesh (2006, 2007, 2013). Lacour et al. (1994) presented a theory to study the effects of compressibility and permeability of the viscous fluid saturating the porous piezoelectric materials on the piezoelectric properties of piezoelectric ceramics. Gomez and Montero (1996b, 1997a,b) dealt with a new system of constitutive relations that describe the elastic, dielectric and the piezoelectric behavior of porous piezoelectric materials. The influence of the coupling mechanisms on the material parameters was also analyzed. Craciun et al. (1998) made an experimental study on wave propagation in porous piezoelectric ceramics in order to improve their properties over dense ceramics. The effects of porosity on phase velocity and attenuation of longitudinal waves were
97
also studied experimentally. Gomez et al. (2000) adopted a 2D model to study the propagation of acoustic plane waves through composite materials using the finite-element method. Altay and Dokmeci (2005) expressed the governing equations of a porous piezoelectric continuum in variational form and obtained a three field variational principle with some conditions. A survey of the literature related to porous piezoelectric materials reveals that very few authors have established theoretical models for porous piezoelectric materials. Vashishth and Gupta (2009a) derived the constitutive equations and equations of motion for porous piezoelectric materials. In order to improve the properties of porous piezoelectric materials, the knowledge of their physical properties and wave phenomena in such a medium is desirable. An analytical study of wave propagation in transversely isotropic porous piezoelectric materials was done by them (Vashishth and Gupta, 2009b). The effects of porosity, direction of propagation and piezoelectricity on the phase velocities, slowness and attenuation coefficients were investigated therein. Sharma (2010) analyzed the effects of piezoelectricity on the phase velocities and group velocities of waves propagating in an anisotropic porous piezoelectric material. Subsequently, the uniqueness theorem, theorem of reciprocity and general theorems in the linear theory of porous piezoelectricity were established (Vashishth and Gupta, 2011a). Based on the theoretical formulation developed, Vashishth and Gupta, 2011b, 2012, 2013 carried out studies of reflection and transmission of waves in models involving porous piezoelectric materials. Three-dimensional wave propagation in porous piezoelectric materials, for different symmetry classes, is studied in this paper. The porous piezoelectric material is assumed to be saturated with a viscous fluid. The Christoffel equation, corresponding to an anisotropic porous piezoelectric medium, is derived. The wave velocities of four inhomogeneous waves are obtained. These wave velocities give further the phase velocities and attenuation coefficients of the corresponding waves. The results for different crystal classes, viz. monoclinic (2, m), orthorhombic (222, 2 mm), tetragonal (4), trigonal (32), hexagonal (6 mm) and cubic (43mÞ are deduced. The effects of phase direction, frequency, piezoelectricity, porosity and crystal symmetry on the velocity surfaces and attenuation surfaces are investigated. The effects of phase direction and crystal symmetry on the skewing angles and wave fronts are also studied. 2. Christoffel equation for anisotropic porous piezoelectric materials A porous piezoelectric material saturated with a viscous fluid is considered. The constitutive equations (Vashishth and Gupta, 2009a) for anisotropic porous piezoelectric materials are
r ¼ c:e þ me e:E f:E ;
ð1Þ
r ¼ m:e þ Re ~f:E e :E ;
ð2Þ
98
A.K. Vashishth, V. Gupta / Mechanics of Materials 80 (2015) 96–112
D ¼ e:e þ ~fe þ n:E þ A:E ;
ð3Þ
D ¼ f:e þ e e þ A:E þ n :E ;
ð4Þ
where rðr Þ and eðe Þ are the stress and strain tensors of order 2 acting on solid (fluid) phase of porous aggregate. EðE Þ and DðD Þ are electric field and electric displacement vectors. c is the elastic stiffness tensor of order 4. The elastic constant R measures the pressure to be exerted on fluid to push its unit volume into the porous matrix. eðe Þ and nðn Þ are piezoelectric and dielectric tensors are of order 3(1) and 2, respectively. The tensors m; f; ~f; and A of order 2; 3,1; and 2 link the elastic; piezoelectric; and dielectric properties of two phases of the porous aggregate. According to Biot, 1956a,b theory, homogeneous porous solid matrix and its saturating fluid are treated in the manner of two interpenetrating elastic continua. Losses arise due to the viscous motion of the fluid relative to the solid matrix. The equations of motion and Gauss equations for such a material are
r:r ¼ q11 :u þ q12 :u þ b:ðu_ u_ Þ;
ð5Þ
r:r ¼ q12 :u þ q22 :u b:ðu_ u_ Þ;
ð6Þ
r:D ¼ 0;
ð7Þ
r:D ¼ 0;
ð8Þ
where uðu Þ is the mechanical displacement vector. UðU Þ is the electric potential. q11 ; q12 and q22 are dynamical tensors of order 2 related to the porous aggregate density (q), pore fluid density (qf Þ and the inertial coupling between the phases. In the low frequency range, the dissipation tensor (b) of order 2 for an anisotropic porous medium is related to porosity (f), fluid viscosity (l) and solid-matrix permeability (v) as
b¼f
2
lv1 :
ð9Þ
In case of high frequency, the friction force of the fluid on the solid becomes out of phase with relative rate of flow and exhibit’s frequency dependence. In high frequency range, l is replaced by lFðjÞ, where the complex function FðjÞ may be defined as (Badiey and Yamamoto, 1985)
FðjÞ ¼
pffiffiffiffiffiffi pffiffiffiffiffiffi ij tanh ij 1 pffiffiffiffiffiffi pffiffiffiffiffiffi ; 3 1 tanh ij = ij
ð10Þ
pffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where j ¼ 6p v0 tqf =lf ; v0 is the norm of the permeability matrix and t is the wave frequency. For plane harmonic waves, the solution of the system (5)–(8) can be considered as
ðu; u ; U; U Þ ¼ ðB; F; G; HÞ exp
n
ix
v
:x t ;
ð11Þ
where i ¼ 1. x is the circular frequency of harmonic waves and v is the wave velocity in the medium. The direction of phase propagation (n) of a wave can be written as
where h is the angle made by the direction of phase propagation with the x3 axis and / is the azimuth angle. Making use of the constitutive Eqs. (1)–(4) and the Eq. (11) in the Eqs. (5)–(8), one can obtain
11 ÞB þ ðK2 h q 12 ÞF ¼ 0; ðK1 h q
ð13Þ
12 ÞB þ ðK4 h q 22 ÞF ¼ 0; ðK3 h q
ð14Þ
where
B ¼ ½B1 B2 B3 T ; F ¼ ½F 1 F 2 F 3 T ;
0
K1 ¼ P þ e0 r0 þ f0 b ;
K2 ¼ ðm:nÞ n þ e0 t0 þ f0 z0 ; 0
K3 ¼ ðm:nÞ n þ n0 r0 þ n b ; K4 ¼ k4 n n; h ¼ 1=v2 ; Pik ¼ cijkl nl nj ; e0k ¼ eikl ni nl ; f0k ¼ fikl ni nl ; n0 ¼ ð~f:nÞn; n ¼ ðe :nÞn; n0 ¼ n:ðn nÞ; n0 ¼ n :ðn nÞ; a ¼ A:ðn nÞ; f0 n0 n0 f0 0 ; 2 0 0 e a a nn a n n0 n0 n 0 ; t0 ¼ 2 0 0 n a a nn a
r0 ¼
0
b ¼
a n0 f0 a n0 n 0 0 0 ; z ; e ¼ n a a a2 n0 n0 a2 n0 n0
k4 ¼ R þ ~f:n½e :n=a n0 ð~f:n e :nn0 =aÞ=ða2 n0 n0 Þ þ e :nað~f:n e :nn0 =aÞ=ða2 n0 n0 Þ; q q b ¼ 22 ¼ q22 þ b; 12 ¼ q12 b; q 11 ¼ q11 þ b;
i b: x
The system (13) and (14) can be written as
CB ¼ 0;
ð15Þ
where
C ¼ B0 h þ C0 þ
D0 ; c1 h þ c2
ð16Þ
X2 1 c2 11 ; C0 ¼ q X3 X2 ; c1 c1 c1 c2 c2 D0 ¼ X4 þ X3 X2 ; c1 c1 B0 ¼ K1
12 þ K2 G0 K3 q 12 F0 K3 ; X2 ¼ K2 F0 q 12 þ q 12 q 12 G0 K3 ; 12 F0 q X3 ¼ K2 G0 q
2
n ¼ ðsin h cos /; sin h sin /; cos hÞ;
ð12Þ
12 ; 12 G0 q X4 ¼ q c1 ¼ k4 n21 d1 þ k4 n22 d4 þ k4 n23 d6 þ 2n1 n2 k4 d2 þ 2n2 n3 k4 d5 þ 2n1 n3 k4 d3 ;
A.K. Vashishth, V. Gupta / Mechanics of Materials 80 (2015) 96–112 0 22 22 22 22 c2 ¼ ðq 22 d4 þ q12 d2 þ q23 d5 Þ; G ¼ adjðq Þ; 2
22 22 22 d1 ¼ q 22 q33 ðq23 Þ ;
as stiffened qS1 and qS2 waves respectively. Correspond1 1 1 ingly, Q 1 represent attenuation 1 ; Q 2 ; Q 3 and Q 4 coefficients of qP 1 ; qP 2 ; qS1 and qS2 waves respectively.
22 22 22 22 d2 ¼ q 13 q23 q12 q33 ;
22 22 22 22 d3 ¼ q 12 q23 q13 q22 ; 2
22 22 22 d4 ¼ q 11 q33 ðq13 Þ ;
3. Numerical results and discussion Numerical computation of the analytical expressions of phase velocities, attenuation coefficients and skewing angles of stiffened qP 1 ; qP 2 ; qS1 and qS2 waves was done for particular models whose data is listed in the Tables 1–6.
22 22 22 22 d5 ¼ q 12 q13 q11 q23 ;
2
22 22 22 d6 ¼ q 11 q22 ðq12 Þ ; 2 22 22 2 22 F 011 ¼ ðq 33 n2 q22 n3 þ 2n2 n3 q23 Þk4 ; 2 22 22 22 22 F 012 ¼ ðq 12 n3 þ q33 n1 n2 q13 n2 n3 q23 n1 n3 Þk4 ;
F 013 F 022
¼
3.1. Skewing angles
2 22 22 22 ð 22 13 n2 23 n1 n2 12 n2 n3 þ 22 n1 n3 Þk4 ; 2 22 2 22 ð 22 33 n1 11 n3 þ 2 13 n1 n3 Þk4 ;
¼ q
q
q
q
q q
q
2 22 22 22 22 F 023 ¼ ðq 23 n1 q13 n1 n2 þ q11 n2 n3 q12 n1 n3 Þk4 ; 2 22 22 2 22 F 033 ¼ ðq 22 n1 q11 n2 þ 2q12 n1 n2 Þk4 :
The condition of existence of non-trivial solution of the system (15) leads to 4
3
2
T 1 h þ T 2 h þ T 3 h þ T 4 h þ T 5 ¼ 0;
ð17Þ
where T 1 ; T 2 ; T 3 ; T 4 and T 5 are given in the Appendix A). The coefficients of the Eq. (17) and other implicit expressions within these coefficients for the monoclinic (2, m), orthorhombic (222, 2 mm), tetragonal (4), trigonal (32), mÞ crystal classes can be hexagonal (6 mm) and cubic (43 deduced from the corresponding expressions for anisotropic case and these are detailed in the Appendix B). The solution of the Eq. (17) gives the four complex roots hj ; ðj ¼ 1; 2; 3; 4Þ and correspondingly wave velocities vj of four waves are obtained. The phase velocities (Vj ; j ¼ 1; 2; 3; 4Þ and attenuation coefficients (Q 1 j ; j ¼ 1; 2; 3; 4Þ for these waves are defined as
Vj ¼
ðReðvj ÞÞ2 þ ðImðvj ÞÞ2 ; Reðvj Þ
Q 1 ¼ 2Imðvj Þ=Reðvj Þ: j
ð18Þ
ð19Þ
Corresponding to the each root hj ; ðj ¼ 1; 2; 3; 4Þ, the homogeneous system (15) is solved for a non-trivial ðjÞ ðjÞ ðjÞ solution ðB1 ; B2 ; B3 Þ, which defines the polarization direction of the jth wave propagating in a porous piezoelectric material. The skewing (deviation) angle of the polarization direction of jth wave from the phase propagation direction is defined as
cj ¼ cos1 ðBðjÞ :nÞ:
99
ð20Þ
Thus, for a given phase direction, four plane harmonic waves propagate in an anisotropic porous piezoelectric material. In general, the polarization vectors of these waves are neither perpendicular nor parallel to the phase direction, so these waves are termed as quasi waves. The wave with the largest phase velocity V1 is termed as stiffened qP 1 wave and the wave with the smallest phase velocity V2 is termed as stiffened qP2 wave. The other two waves, with phase velocities V3 and V4 , are named
The range of skewing angles c1 ; c2 ; c3 and c4 for all the considered crystal classes is shown in the Table 7. It is clear from the Table that, the skewing angles associated with the qP1 and qP 2 waves are not equal to 0° and that associated with qS1 and qS2 waves are not equal to 90° , so these waves are termed as quasi–longitudinal and quasi-shear waves respectively. The smallest skewing angle is associated with the fastest qP1 wave. In case of 6 mm class, qS2 wave is a pure transverse wave in all the directions of propagation which validates the present model’s results for hexagonal symmetry. In the crystal class 43m, the skewing angles associated with qP1 and qP2 waves are near to 0° and those corresponding to qS1 and qS2 are near to 90° meaning thereby that qP 1 and qP 2 are nearly longitudinal and qS1 and qS1 are nearly transverse waves. 3.2. Phase velocities The phase velocities of qP 1 ; qP 2 ; qS1 and qS2 waves were computed in low frequency range (LFR) and high frequency range (HFR). The value of t is considered as 50 Hz in LFR and 1 MHz in HFR. Fig. 1(a) and (b) exhibit the variations of phase velocities of the four waves in porous piezoelectric materials, saturated with viscous fluid, for the crystal classes m and 2, respectively with phase direction. The elastic, dielectric and dynamical constants of these two classes are same but piezoelectric constants are different. Elevations and depressions of phase velocity surfaces from the horizontal plane in any plot represent the velocity anisotropy for the corresponding wave in the medium. The effects of azimuth variations on quasi shear waves are more prominent in comparison to the quasi longitudinal waves. The phase velocity of qP 1 wave increases with h but the phase velocity of qP 2 wave decreases with an increase in h . The phase velocity of qS1 wave first decreases with an increase in h and attains a local minimum at h ¼ 30 but, after that, it increases with h. Contrary to this, V4 increases first and then decreases having a minimum at 45° and then increases further. The difference in the velocities of qS1 and qS2 waves reveals the anisotropy effect of monoclinic symmetry. Comparison of the respective graphs in Fig. 1(a) and (b) reveals that the phase velocities of qP 1 ; qS1 and qS2 waves in both the classes (m and 2) of monoclinic symmetry are not significantly different. The variation pattern of phase velocity of qP 2 wave also remains same but its magnitude for class m is less than that for class 2. However, such a phenomenon
100 Table 1 Density q
A.K. Vashishth, V. Gupta / Mechanics of Materials 80 (2015) 96–112
ðKg=m3 Þ, elastic stiffness coefficients cIJ ðGPaÞ and elastic coupling coefficients mij ðGPaÞ for different crystal classes (Auld, 1973).
Crystal Class
Material
q
c11
c12
c13
c16
c22
c23
c26
c33
c36
c44
c45
c55
c66
m11
m12
m22
m33
2, m
Barium Sodium Niobate Rochelle Salt Barium Sodium Niobate Barium Titanate Barium Titanate Bismuth Germanate
5300
239
104
50
4.2
247
52
5.7
135
3.2
65
5.4
66
76
8.8
11.5
16.8
5.2
1767
28
17.4
15
0
41.4
19.7
0
39.4
0
6.66
0
2.85
9.6
8.8
0
16.8
5.2
5300
239
104
50
0
247
52
0
135
0
65
0
66
76
8.8
0
16.8
5.2
6020
275
179
151
50
275
151
50
165
0
54.3
0
54.3
113
8.8
0
8.8
5.2
5700
150
66
66
0
150
66
0
146
0
44
0
44
42
8.8
0
8.8
5.2
7095
115.8
27
27
0
115.8
27
0
115.8
0
43.6
0
43.6
43.6
8.8
0
8.8
8.8
222 2 mm
4 6 mm m 43
Table 2 Density q
ðKg=m3 Þ, elastic stiffness coefficients cIJ ðGPaÞ and elastic coupling coefficients mij ðGPaÞ for crystal class 32 (Auld, 1973).
Crystal Class
Material
q
c11
c12
c13
c14
c15
c24
c25
c33
c44
m11
m33
32
Quartz
2651
86.74
6.99
11.91
17.91
0
17.91
0
107.2
57.94
8.8
5.2
Table 3 Piezoelectric constants eiJ , ei and ~fi for different crystal classes; units = C=m2 (Auld, 1973). Crystal Class e14
e15
e24
e25
e31
e32
e33
e36
e3
f14
f15
f24
f25
f31
f32
f33
f36
~f3
2 222 2 mm 4 6 mm m 43
2.8 0 2.8 21.3 11.4 0
3.4 0 3.4 21.3 11.4 0
4.2 0.154 0 15 0 0.038
0.4 0 0.4 2.74 4.35 0
0.3 0 0.3 2.74 4.35 0
4.3 0 4.3 3.7 17.5 0
5.6 0.115 0 0 0 0.038
0.6 0 0.6 0.6 0.6 0
0.12 0.06 0 0.375 0 .001
0.07 0 0 0.53 0.285 0
0.09 0 0.085 0.53 0.285 0
0.11 .0004 0 0.38 0 0.001
0.01 0 0.01 0.07 0.11 0
0.008 0 0.008 0.069 0.11 0
0.108 0 0.108 0.093 0.093 0
0.14 0.003 0.003 0 0 0.001
0.75 0 0.75 0.75 0.75 0
4.8 2.23 0 15 0 0.038
Table 4 Piezoelectric constants eiJ , ei and ~fi for the crystal classes m and 32; units = C=m2 (Auld, 1973). Crystal Class
Piezoelectric constants
m
e11 e23 f11 f23
32
e11 ¼ 0:171
¼ 0:2 ¼ 0:46 ¼ 0:005 ¼ 0:012
e12 e26 f12 f26
¼ 0:07 ¼ 0:89 ¼ 0:0018 ¼ 0:022
e13 e34 f13 f34
e14 ¼ 0:0436
¼ 0:32 ¼ 0:12 ¼ 0:008 ¼ 0:003
e3 ¼ 0:6
¼ 0:067 ¼ 0:67 ¼ 0:0017 ¼ 0:017
f11 ¼ 0:004
Table 5 Dielectric constants nij ðnC=VmÞ; nij ðnC=VmÞ and Aij ðnC=VmÞ for different crystal classes,
2, m 222 2 mm 4 32 6 mm m 43
e16 e35 f16 f35
e21 ¼ 0:4 e1 ¼ 2:6 f21 ¼ 0:01 ~f1 ¼ 1:2
e22 ¼ 0:23 e2 ¼ 1:7 f22 ¼ 0:0058 ~f2 ¼ 0:4
f14 ¼ 0:0011
~f3 ¼ 0:4
e0 ¼ 8:854 1012 is the absolute permittivity (Auld, 1973).
n11 =e0
n12 =e0
n22 =e0
n33 =e0
n11
n12
n22
n33
A11
A12
A21
A22
A33
1220 205 2.35 2920 4.52 1450 16
1107 0 0 0 0 0 0
1672 9.6 247 2920 4.52 1450 16
1480 9.5 51 168 4.68 1700 16
0:038 0:038 0:038 0.038 0.038 0.038 0.038
0:09 0 0 0 0 0 0
0:055 0:055 0:055 0.038 0.038 0.038 0.038
0:049 0:049 0:049 0.049 0.049 0.049 0.038
0:018 0:018 0:018 0.018 0.018 0.018 0.018
0:04 0 0 0.04 0 0 0
0:08 0 0 0.04 0 0 0
0:031 0:031 0:031 0.018 0.018 0.018 0.018
0:015 0:015 0:015 0.015 0.015 0.015 0.018
101
A.K. Vashishth, V. Gupta / Mechanics of Materials 80 (2015) 96–112 Table 6 Dynamical coefficients and other parameters. R ¼ 20 GPa
v11 ¼ 1:0 10 q11 =q ¼ 0:66
qf ¼ 1000Kg=m3 10
m
10
2
v12 ¼ 0:5 10 q12 =q ¼ 0:15
l ¼ 1 103 Ns=m2 m
10
2
v22 ¼ 1:3 10 q22 =q ¼ 0:64
m
f ¼ 0:2 2
v33 ¼ 0:8 1010 m2 –
Table 7 Range of skewing angle of all the waves in different crystal classes. Crystal class
m 2 222 2 mm 4 32 6 mm m 43
Range of skewing angle qP 1 wave
qP 2 wave
qS1 wave
qS2 wave
0:8 19 0:8 19 0:9 15 0:7 15:3 0:2 16 0:06 21:4 0:002 2:1 0:01 0:32
0:3 14 0:6 30 13 29 1:1 39 9:1 25 0:3 38 6:8 22 0:04 0:97
74:2 89 74:4 89 71 89 79:3 89 74:8 89 69:5 89 87:5 89 89:6 89:9
74 89 74:2 89 77 89 78 89 75:5 89 71:7 89 90 89:8 89:9
does not exist in a poro-elastic medium. The velocities of all the quasi waves increase as the frequency shifts from LFR to HFR. The pattern of variation of phase velocities with the phase directions remains unaffected in either case. Phase velocities variations with directions and with frequency, for the crystal classes 222, 2 mm, 4, 32, 6 mm and m, are shown in Fig. 1(a) and (h). It is observed that the 43 velocities of all the waves increase as the frequency shifts from LFR to HFR in all the crystal classes except the crystal classes 2 mm and 4. The noticed changes in the phase velocities are due to changes in crystal classes and crystal type both. It is interesting to note that the azimuthal variations do considerable affect the phase velocities of all waves in the case of crystal classes 222, 4 and 32 Fig. 1(c), (e) and (f). The azimuthal variation has no effects on velocity pattern in case of 6 mm class, which confirms the results (Fig. 1(g)). Unlike the above mentioned crystal classes, velocity surfaces for LFR and HFR in case of cubic m are found exactly parallel. The effects of polar class 43 angle on the phase velocities of all the waves are significant in all the crystal classes. To study the effects of anisotropy on the variation of phase velocities, the results are deduced numerically for m from the crystal classes 222, 2 mm, 4, 6 mm and 43 the Barium Sodium Niobate crystal of class 2 but the plots m to are shown here only for the classes 222, 4 and 43 avoid the repetition. The magnitude as well as the variation pattern of qP1 wave remains unaffected in all the crystal classes. The velocity of qP 2 wave changes with crystal classes. Noticeable deviation is observed from class 2 to 222 (Fig. 2(a)). Exactly similar variation of V 2 is observed for 2 to 43mclass and so is not shown here. However, the velocity of qP2 wave is found to be same in case of 2, 2 mm, 4 and 6 mm classes. The phase velocities of qP1 ; qS1 and qS2 waves are same for 222 and 2 mm classes. qS1 velocity variations with directions is different for 2 and 222 crystal classes (Fig. 2(a)). The effects of crystal
symmetry on the phase velocity of qS1 wave are more prominently observed in cases of the crystal classes 4 and 43m (Fig. 2(b) and (c)). It is observed that velocity surfaces of all waves coincide for 2 and 6 mm classes. In m, the all the crystal classes, except the crystal class 43 magnitude and variation pattern of phase velocity of qS2 wave also remains same except for small values of polar angle. Thus, it may be concluded that the effects of crystal symmetry on the magnitudes of phase velocities are least in case of fastest waves. Fig. 3 shows the effects of pore volume fraction on the phase velocities of quasi waves in LFR for the crystal class m. The effects of pore volume fraction on the phase velocities are not significant in the HFR. It is observed that in case of LFR, the phase velocities of all the waves decrease with the increase of porosity in all the phase directions. Percolation theory (Craciun et al., 1998) may be able to explain the decrease in phase velocity with an increase of porosity. In this theory, a geometric phase transition occurs in the percolation system when the medium becomes disconnected due to the progressive dilution. The phase velocities of qP 1 ; qS1 and qS2 waves in the LFR do not change with the increase of porosity beyond 0.6 approximately. However, the slowest wave is found to be more sensitive to changes in the porosity of the medium. Thus, it can be concluded that introduction of pores in piezoceramics affects the wave velocities but only up to a certain level of porosity. Similar velocity patterns were also observed for other crystal classes and results for those classes have been found not significantly different and thus are not shown here. The wavefronts (slowness surfaces) were obtained for all different crystal classes but these are shown for the crystal classes m, 222, 4 and 32 in Fig. 4(a) and (d). The acoustic wavefronts play a central role in the analysis of many crystal acoustic phenomena, including phonon focusing, reflection and transmission at surfaces, and surface and interface waves [8]. Any of the two wavefronts of the four
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Fig. 1. Effects of frequency on the variation of phase velocities of quasi waves with phase direction; (a) Barium Sodium Niobate crystal of class m, (b) Barium Sodium Niobate crystal of class 2, (c) Rochelle Salt crystal of class 222, (d) Barium Sodium Niobate crystal of class 2 mm, (e) Barium Titanate crystal m. of class 4, (f) Quartz crystal of class 32, (g) Barium Titanate crystal of class 6 mm, (h) Bismuth Germanate crystal of class 43
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Fig. 1 (continued)
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Fig. 2. Anisotropic variation of phase velocities of quasi waves for Barium Sodium Niobate crystal: Reduced from the crystal class 2; (a) Class 222, (b) Class 4 m. (c) Class 43
Fig. 3. Effects of porosity on the anisotropic variation of phase velocities of quasi waves for Barium Sodium Niobate crystal of class m.
waves may meet in a number of isolated directions. There are three types of singularities (Crampin, 1981): kiss singularities, where the two sheets touch tangentially with either convex or concave contact; intersection singularities,
where the two sheets may be considered as cutting each other along a closed curve; and point singularities, where the two sheets have common vertices of cone-shaped projection of surfaces. The number of singularities as well
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Fig. 4. Wavefront of quasi waves; (a) Barium sodium Niobate crystal of class m, (b) Rochelle salt crystal of class 222, (c) Barium Titanate crystal of class 4, (d) Quartz crystal of class 32.
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Table 8 Two-shear wavefronts singularities directions in different crystal classes. Crystal Class
Two-shear wavefronts singularities directions
m 2 2 mm 4
ð29 ; 67 Þ; ð30 ; 66 Þ & ð30 ; 67 Þ ð29 ; 65 Þ; ð29 ; 66 Þ & ð29 ; 67 Þ h ¼ 29 ; 84 6 / 6 90 ; h ¼ 30 ; 84 6 / 6 90 h ¼ 0 ; 0 6 / 6 90 ; h ¼ 1 ; 0 6 / 6 90 ; h ¼ 2 ; 34 6 / 6 52 h ¼ 0 ; 0 6 / 6 90 & ð67 ; 30 Þ 0 6 h 6 20 ; 0 6 / 6 90
32 6 mm
is summarized in the Table 8. In case of the crystal classes 222 and 4, the wavefronts of qP2 and qS2 waves also have common points of contact. In case of the crystal class 222, qP2 and qS2 waves have common phase velocities in the direction ð68 ; 80 Þ while in case of the crystal class 4, these two waves have common phase velocities in the directions h ¼ 35 ; / ¼ 74 76 ; h ¼ 36 ; / ¼ 70 72 and h ¼ 37 ; / ¼ 68 69 . 3.3. Attenuation coefficients
as their types may vary due to changes in crystal symmetry. qS1 and qS2 waves have common phase velocities in different phase directions in different crystal classes, which
The attenuations of qP1 ; qP 2 ; qS1 and qS2 waves in PPM saturated with viscous fluid are computed for different crystal classes in the LFR and HFR. In the LFR, the attenuation is
Fig. 5. Variation of attenuation coefficients with the pore volume fraction for Barium Sodium Niobate crystal of class m; (i) LFR (ii) HFR.
Fig. 6. Variation of phase velocities of quasi waves with phase direction; porous non-piezoelectric materials.
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Fig. 7. Effects of piezoelectricity on the variation of phase velocities of quasi waves with phase direction; (a) Barium Sodium Niobate crystal of class 2, (b) Rochelle salt crystal of class 222, (c) Barium Sodium Niobate crystal of class 2 mm, (d) Barium Titanate crystal of class 4, (e) Barium Titanate crystal of class 6 mm.
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Fig. 7 (continued)
mainly contributed by viscosity losses due to the friction at the fluid-grain boundary. The attenuation of the slowest wave is maximum while that of the fastest wave is minimum. It is observed that the variation pattern of attenuations of qP1 and qP2 waves with h is same as for phase velocities while attenuations of qS1 and qS2 waves behave opposite to phase velocities variation and thus plots are omitted here. It is also observed that unlike the phase velocities, the attenuations of all the waves decrease as the frequency shifts from LFR to HFR. The effects of azimuth variation on the attenuation coefficient of qP1 wave are more significant for high value of h while these effects are found significant for small values of h in case of qS1 and qS2 waves. The effects of electro-elastic interaction on the attenuation coefficients in the HFR for different crystal classes are studied. It is found that the electro-elastic interaction has very small effects on the magnitude of attenuation coefficients and pattern of variation thereof in all the crystal classes except the crystal classes 4 and 6 mm. In case of classes 4 and 6 mm, the attenuations of qP 1 ; qS1 and qS2 waves increase due to inclusion of piezoelectric effect like the phase velocities while that of qP 2 wave decreases due to these effects unlike the phase velocity and thus the plots are not shown here. Except qP1 wave, the electro-elastic interaction has significant effects on the pattern of variation of attenuation coefficients of all the waves with the phase direction. Thus, it can be inferred that the effects of electro-elastic interaction on the attenuation of waves are different in different materials having different kinds of symmetry. Fig. 5(i) and (ii) show the dependence of the attenuation coefficients on the pore volume fraction for the crystal class m in the LFR and HFR, respectively. The value of h and / are considered as 30 and 60 , respectively. It is observed that in the LFR, the attenuation of qP 1 ; qS1 and qS2 waves first increase with porosity and attain local maxima at 35% porosity and after that it decreases monotonically with an increase in porosity. However, the attenuation of qP2 wave increases monotonically with porosity and its attenuation is the maximum of all the waves in LFR, which is so. However, in the HFR, the attenuation of qP2 wave is comparable to the other waves, which is in conformity with the fact that in the high frequency range
qP2 wave is propagating in nature. The attenuation of all the waves increases rapidly with porosity in the HFR. Similar results were also observed for other crystal classes and results for those classes have been found not significantly different and thus are not shown here. 4. Particular cases In order to validate the considered model of study here, the earlier established results are obtained. as particular cases of the present model, which are described as follows: 4.1. 3-D waves in porous non-piezoelectric materials The correspondence between porous piezoelectric materials and porous non-piezoelectric materials can be obtained by substitution of e; f; e ; ~f; n; n ; A; D; D ; E; E ; G; H as zero. The homogeneous system for porous nonpiezoelectric materials can be obtained from the homogeneous system (15), derived for porous piezoelectric materials, by substitution of e; f; e ; ~f; n; n ; A; D; D ; E; E ; G; H as zero. The condition of existence of non-trivial solution of the homogeneous system, corresponding to porous non-piezoelectric material case, also leads to biquadratic equation in h. Thus, the number of waves remains unaffected due to the effects of piezoelectricity. The phase velocities of all the quasi waves are obtained for porous non-piezoelectric materials by neglecting effects of piezoelectricity and considering the data for elastic constants and other parameters listed in Sharma (2004). The variation of phase velocities of waves propagating in porous non-piezoelectric materials with the direction of propagation is shown in Fig. 6 and the results are found to be in agreement with the results of Sharma (2004). Further, to study the effects of piezoelectricity on the magnitude of phase velocities of quasi waves, the results are computed for porous non-piezoelectric materials (PM) and porous piezoelectric materials (PPM) cases. A magnificent effects of piezoelectricity are not observed in m and thus are the case of crystal classes m, 32 and 43 not shown here. The effects of piezoelectricity on the variation of phase velocities with the phase directions for
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Fig. 8. Variation of phase velocities of quasi waves with phase direction; non-porous piezoelectric material; Barium Sodium Niobate crystal of class m.
Fig. 9. Slowness surfaces cut by the x1 x2 plane; non-porous piezoelectric materials; Lithium Niobate crystal of class 3 m.
the crystal classes 2, 222, 2 mm, 4 and 6 mm are shown in Fig. 7(a) and (e). For the classes 2 and 4, electro-elastic interaction has weak effects on qP 1 ; qS1 and qS2 waves but phase velocity of qP 2 wave increases due to piezoelectricity (Fig. 7(a) and (d)). The magnitudes of phase velocities of all the waves remain almost unaffected due to piezoelectricity while the variation pattern of qS2 wave gets effected due to electro-elastic interaction in the crystal class 222 (Fig. 7(b)). In case of class 2 mm (Fig. 7(c)), the phase velocities of qP 1 ; qP 2 and qS2 waves increase due to piezoelectricity while that of qS1 wave remains unaffected. The velocity of qP1 wave increases due to piezoelectricity only for small values of polar angle while that of qS2 wave increases due to piezoelectricity except for small values of polar angle. The velocity of the slowest wave increases for all values of polar angle. Fig. 7(e) exhibits that the phase velocities of all the quasi waves increase due to electroelastic interaction in case of 6 mm class. The effects of piezoelectricity on the fastest wave are more significant in this case. Thus, it may be concluded that the different crystals having a different piezoelectric coupling have different wave velocities and consideration of piezoelectricity in the wave phenomenon is important for some of the crystal classes. It is also justified from here that Barium Titanate (6 mm) has a strong piezoelectric coupling in comparison to Quartz crystal (32). These observations are found in agreement with the experimental ones (Jaffe et al., 1971).
4.2. 3-D waves in non-porous piezoelectric materials To study the qualitative difference between 3-D waves in non-porous piezoelectric materials and porous piezoelectric materials, the homogeneous system (15), derived for porous piezoelectric materials, is reduced for non-porous piezoelectric materials by substitution of m; R; f; e ; ~f; n ; A; q12 ; q22 ; b; F; H; r ; e ; u ; E ; D as zero which further leads to a cubic equation in h unlike the case of porous piezoelectric materials. Thus, the number of waves propagating in non-porous piezoelectric materials is three unlike the PPM case. It is also found that the waves propagating in non-porous piezoelectric material are mutually orthogonal but the waves are not mutually orthogonal in case of porous piezoelectric materials. The variation of phase velocities of waves, propagating in non-porous piezoelectric materials having crystal symmetry monoclinic (m), with direction of propagation is shown in Fig. 8. The results for non-porous piezoelectric materials having crystal symmetry trigonal (3 m) have been obtained as a particular case so that present results are validated (Fig. 9). The elastic, piezoelectric and dielectric constants listed in the text Auld (1973) are considered. The variation of slowness with the direction of propagation in the x1 x2 plane is found to be in agreement with the results of Auld (1973), as should be.
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5. Conclusion The three-dimensional wave propagation in porous piezoelectric materials saturated with viscous fluid is studied. The Christoffel equation is derived for anisotropic porous piezoelectric materials and the algebraic expressions of the Christoffel coefficients are deduced for crystal m. The effects classes 2, m, 222, 2 mm, 4, 32, 6 mm and 43 of frequency, porosity, piezoelectricity and crystal symmetry on the velocity surfaces and attenuation surfaces of four waves propagating in a porous piezoelectric medium are studied numerically for particular models. The skewing angles associated with the quasi longitudinal waves are smaller in comparison to the angles associated with the quasi-shear waves. In case of 6 mm class, qS2 wave is a pure transverse wave in all the directions of propagation. The effects of polar angle and azimuth angle on the phase velocities of waves are different for different crystal classes. The effects of azimuth variations on quasi shear waves are more prominent in comparison to quasi longitudinal waves. The velocity of the slowest wave for the class m is 16% less than in comparison to the class 2 while velocities of other three waves remain same in both the classes of a monoclinic group. The azimuth variation has no effects on the phase velocities in the case of the crystal class 6 mm. The effects of crystal symmetry on the magnitudes of phase velocities are least in case of fastest waves while these effects on the other three waves are different for different crystal classes. The velocities of all quasi waves increase while attenuations decrease as the frequency shifts from LFR to HFR. The phase velocities of all waves decrease with an increase in porosity in LFR. The slowest wave is found to be more sensitive to the porosity of the medium. The introduction of pores in piezoceramics affects the wave velocities but only up to a certain level. The slowest wave is highly attenuated while the fastest wave is least attenuated. The changes in the attenuation of waves are less significant due to change of crystal symmetry. The attenuation of all the waves increases with porosity in HFR. Different crystals having a different piezoelectric coupling have different wave velocities and attenuations and consideration of piezoelectricity for wave phenomenon is important for some of the crystal classes. Barium Titanate has a strong piezoelectric coupling in comparison to Quartz crystal. The wavefronts of two quasi-shear waves and slowest quasi-shear and slowest quasi-longitudinal waves may coincide along some phase directions, and these directions get affected due to crystal symmetry. The results are obtained for the particular cases porous non-piezoelectric materials and non-porous piezoelectric materials and are found in agreement with the earlier established results. Appendix A
T 01 ¼ B011 y1 þ B012 y6 þ B013 y11 ; T 02 ¼ C 011 y1 þ C 012 y6 þ C 013 y11 þ B011 y2 þ B012 y7 þ B013 y12 ;
T 03 ¼ C 011 y2 þ C 012 y7 þ C 013 y12 þ B011 y4 þ B012 y9 þ B013 y14 ; T 04 ¼ C 011 y4 þ C 012 y9 þ C 013 y14 ;
T 05 ¼ D011 y1 þ D012 y6 þ D013 y11 þ B011 y3 þ B012 y8 þ B013 y13 ;
T 06 ¼ D011 y2 þ D012 y7 þ D013 y12 þ C 011 y3 þ C 012 y8 þ C 013 y13 þ B011 y5 þ B012 y10 þ B013 y15 ;
T 07 ¼ D011 y4 þ D012 y9 þ D013 y14 þ C 011 y5 þ C 012 y10 þ C 013 y15 ;
y1 ¼ B022 B033 B023 B032 ; y2 ¼ B022 C 033 þ C 022 B033 B023 C 032 C 023 B032 ;
y3 ¼ B022 D033 þ D022 B033 B023 D032 D023 B032 ; y4 ¼ C 022 C 033 C 023 C 032 ;
y5 ¼ C 022 D033 þ D022 C 033 C 023 D032 D023 C 032 ; y6 ¼ B031 B023 B021 B033 ;
y7 ¼ B031 C 023 þ C 031 B023 B021 C 033 C 021 B033 ; y8 ¼ B021 D033 D021 B033 þ B023 D031 þ D023 B031 ;
y9 ¼ C 021 C 033 þ C 023 C 031 ; y10 ¼ C 021 D033 D021 C 033 þ C 023 D031 þ D023 C 031 ;
y11 ¼ B021 B032 B031 B022 ; y12 ¼ B021 C 032 þ C 021 B032 B031 C 022 C 031 B022 ;
y13 ¼ B021 D032 þ D021 B032 B031 D022 D031 B022 ; y14 ¼ C 021 C 032 C 022 C 031 ;
T 1 ¼ c1 T 01 ; T 2 ¼ c2 T 01 þ c1 T 02 ; T 3 ¼ c2 T 02 þ c1 T 03 þ T 05 ; T 4 ¼ c2 T 03 þ c1 T 04 þ T 06 ; T 5 ¼ c2 T 04 þ T 07
y15 ¼ C 021 D032 þ D021 C 032 C 022 D031 D022 C 031 :
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Appendix B 2
e01 ¼ ðe15 þ e31 Þn1 n3 þ ðe25 þ e36 Þn2 n3 ; e02 ¼ ðe14 þ e36 Þn1 n3 þ ðe24 þ e32 Þn2 n3 ; e03 ¼ e15 n21 þ e24 n22 þ e33 n23 þ ðe14 þ e25 Þn1 n2 ; nk ¼ e3 n3 nk :
m
e01 ¼ e11 n21 þ e26 n22 þ e35 n23 þ ðe16 þ e21 Þn1 n2 ; e02 ¼ e16 n21 þ e22 n22 þ e34 n23 þ ðe12 þ e26 Þn1 n2 ; e03 ¼ ðe13 þ e35 Þn1 n3 þ ðe23 þ e34 Þn2 n3 ; nk ¼ ðe1 n1 þ e2 n2 Þnk .
2, m
P 11 ¼ c11 n21 þ 2c16 n1 n2 þ c66 n22 þ c55 n23 ; P12 ¼ c16 n21 þ ðc12 þ c66 Þn1 n2 þ c26 n22 þ c45 n23 ; P 13 ¼ ðc13 þ c55 Þn1 n3 þ ðc36 þ c45 Þn2 n3 ; P22 ¼ c66 n21 þ 2c26 n1 n2 þ c22 n22 þ c44 n23 ; P 23 ¼ ðc36 þ c45 Þn1 n3 þ ðc23 þ c44 Þn2 n3 ; P33 ¼ c55 n21 þ 2c45 n1 n2 þ c44 n22 þ c33 n23 ; n0 ¼ n11 n21 þ n22 n22 þ n33 n23 þ 2n12 n1 n2 ; n0 ¼ n11 n21 þ n22 n22 þ n33 n23 þ 2n12 n1 n2 ; a ¼ A11 n21 þ A22 n22 þ A33 n23 þ 2A12 n1 n2 ; c1 ¼ k4 n21 d1 þ k4 n22 d4 þ k4 n23 d6 þ 2n1 n2 k4 d2 ; 2
0 22 22 22 22 22 22 22 22 2 22 c2 ¼ ðq 12 d2 þ q22 d4 Þ; d2 ¼ q12 q33 ; d6 ¼ q11 q22 ðq12 Þ ; F 12 ¼ ðq12 n3 þ q33 n1 n2 Þk4 ; 0 0 0 22 22 22 22 2 22 22 2 22 F 13 ¼ ðq12 n2 n3 þ q22 n1 n3 Þk4 ; F 23 ¼ ðq11 n2 n3 q12 n1 n3 Þk4 ; F 33 ¼ ðq 22 n1 q11 n2 þ 2q12 n1 n2 Þk4 .
2 mm 222 2 mm, 222
e01 ¼ ðe15 þ e31 Þn1 n3 ; e02 ¼ ðe24 þ e32 Þn2 n3 ; e03 ¼ e15 n21 þ e24 n22 þ e33 n23 ; nk ¼ e3 n3 nk , e01 ¼ ðe25 þ e36 Þn2 n3 ; e02 ¼ ðe14 þ e36 Þn1 n3 ; e03 ¼ ðe25 þ e14 Þn1 n2 ; nk ¼ 0. P 11 ¼ c11 n21 þ c66 n22 þ c55 n23 ; P 12 ¼ ðc12 þ c66 Þn1 n2 ; P13 ¼ ðc13 þ c55 Þn1 n3 ; P 22 ¼ c66 n21 þ c22 n22 þ c44 n23 ; P 23 ¼ ðc23 þ c44 Þn2 n3 ; P33 ¼ c55 n21 þ c44 n22 þ c33 n23 ; n0 ¼ n11 n21 þ n22 n22 þ n33 n23 ; n0 ¼ n11 n21 þ n22 n22 þ n33 n23 ; a ¼ A11 n21 þ A22 n22 þ A33 n23 ; c1 ¼ k4 n21 d1 þ k4 n22 d4 þ k4 n23 d6 ; 22 22 22 0 22 c2 ¼ q 22 d4 ; d2 ¼ 0; d6 ¼ q11 q22 ; F 12 ¼ q33 n1 n2 k4 ;
2, 222, m, 2 mm 4
0 0 22 22 22 2 22 2 F 013 ¼ q 22 n1 n3 k4 ; F 23 ¼ q11 n2 n3 k4 ; F 33 ¼ ðq22 n1 q11 n2 Þk4 . 0 0 22 22 22 22 2 22 q 33 ; d4 ¼ q 11 q 33 ; d3 ¼ d5 ¼ 0; F 11 ¼ ðq 22 22 2 22 2 22 2 d1 ¼ q 33 n2 q22 n3 Þk4 ; F 22 ¼ ðq33 n1 q11 n3 Þk4 .
P 11 ¼ c11 n21 þ c66 n22 þ c44 n23 þ 2c16 n1 n2 ; P12 ¼ c16 n21 c16 n22 þ ðc12 þ c66 Þn1 n2 ; P 13 ¼ ðc13 þ c44 Þn1 n3 ; P22 ¼ c66 n21 þ c11 n22 þ c44 n23 2c16 n1 n2 ; P 23 ¼ ðc13 þ c44 Þn2 n3 ; P33 ¼ c44 n21 þ c44 n22 þ c33 n23 ; e01 ¼ ðe15 þ e31 Þn1 n3 e14 n2 n3 ; e02 ¼ ðe15 þ e31 Þn2 n3 þ e14 n1 n3 ; e03 ¼ e15 n21 þ e15 n22 þ e33 n23 ; nk ¼ e3 n3 nk .
32
P 11 ¼ c11 n21 þ c66 n22 þ c44 n23 þ 2c14 n2 n3 ; P12 ¼ ðc12 þ c66 Þn1 n2 þ 2c14 n1 n3 ; P 13 ¼ 2c14 n1 n2 þ ðc13 þ c44 Þn1 n3 ; P 22 ¼ c66 n21 þ c11 n22 þ c44 n23 2c14 n2 n3 ; P 23 ¼ c14 n21 c14 n22 þ ðc13 þ c44 Þn2 n3 ; P 33 ¼ c44 n21 þ c44 n22 þ c33 n23 ; e01 ¼ e11 n21 e11 n22 e14 n2 n3 ; e02 ¼ 2e11 n1 n2 þ e14 n1 n3 ; e03 ¼ 0; nk ¼ 0.
Note: In all the crystal classes, the expressions for ~fi ði ¼ 1; 2; 3Þ can be deduced from those of e0i by replacing eiJ ðJ ¼ 1; 2; . . . ; 6Þ with fiJ . Similarly n0k ðk ¼ 1; 2; 3Þ can be deduced from nk by replacing ek with ~fk . For 6 mm class the results are obtained from 2 mm class by equating quantities with subscripts 22, 24, 32, 55 by the quantities with subscripts 11, 15, 31, 44 respectively. For 43m class, the results are obtained from 222 class by equating quantities with subscripts 22, 23, 25, 33, 36, 55, 66 by the quantities with subscripts 11, 13, 14, 11, 14, 44, 44 respectively. The expressions of n0 ; n0 ; a; c1 ; c2 ; di ði ¼ 1; 2; . . . ; 6Þ; F ij ði; j ¼ 1; 2; 3Þ for the crystal classes 4 and 32 are same as of the crystal class 6 mm.
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Gomez, T.E., Montero, F., 1996b. New constitutive relations for piezoelectric composites and porous piezoelectric ceramics. J. Acoust. Soc. Am. 100, 3104–3114. Gomez, T.E., Montero, F., 1997a. Characterization of porous piezoelectric ceramics: the length expander case. J. Acoust. Soc. Am. 102, 3507– 3515. Gomez, T.E., Montero, F., 1997b. Determination of the electromechanical coupling coefficients of porous piezoceramics. IEEE Ultrason. Symp. 2, 943–946. Gomez, T.E., Mulholland, A.J., Hayward, G., Gomatam, J., 2000. Wave propagation in 0–3/3–3 connectivity composites with complex microstructure. Ultrasonics 38, 897–907. Jaffe, B., Cook, W.R., Jaffe, H., 1971. Piezoelectric Ceramics. Academic Press, New York. Kar-Gupta, R., Venkatesh, T.A., 2006. Electromechanical response of porous piezoelectric materials. Acta Mater. 54, 4063–4078. Kar-Gupta, R., Venkatesh, T.A., 2007. Electromechanical response of 1–3 piezoelectric composites: a numerical model to assess the effects of fiber distribution. Acta Mater. 55, 1275–1292. Kar-Gupta, R., Venkatesh, T.A., 2013. Electromechanical response of (2–2) layered piezoelectric composites. Smart Mater. Struct. 22, 1–14, 025035. Kyame, J.J., 1949. Wave propagation in piezoelectric crystals. J. Acoust. Soc. Am. 21, 159–167. Lacour, O., Lagier, M., Sornette, D., 1994. Effect of dynamic fluid compressibility and permeability on porous piezoelectric ceramics. J. Acoust. Soc. Am. 96, 3548–3557. Lee, S.H., Jun, S.H., Kim, H.E., Koh, Y.H., 2008. Piezoelectric properties of PZT-based ceramic with highly aligned pores. J. Am. Ceram. Soc. 91, 1912–1915. Levassort, F., Holc, J., Ringgaard, E., Bove, T., Kosec, M., Lethiecq, M., 2007. Fabrication, modelling and use of porous ceramics for ultrasonic transducer applications. J. Electroceram. 19, 125–137. Matsunaka, T., Tabuchi, Y., Kasai, C., Tachikawa, K., Kyono, H., Ikeda, H., 1988. Porous piezoelectric ceramic transducer for medical ultrasonic applications. Ultrason. Symp., 681–684. Nayfeh, A.H., 1995. Wave Propagation in Layered Anisotropic Media with Applications to Composites. North-Holland. Rao, B.S., 1978. Wave propagation in piezoelectric medium of hexagonal symmetry. Proc. Indian Acad. Sci. 87A, 125–136.
Romeo, M., 1996. Material symmetries and inhomogeneous waves in piezoelectric media. Acta Mech. 118, 27–37. Roncari, E., Galassi, C., Craciun, F., Capiani, C., Piancastelli, A., 2001. A microstructural study of porous piezoelectric ceramics obtained by different methods. J. Eur. Ceram. Soc. 21, 409–417. Royer, D., Dieulesaint, E., 1996. Elastic Waves in Solids I Free and Guided Propagation. Springer. Sharma, M.D., 2004. Wave propagation in a general anisotropic poroelastic medium with anisotropic permeability: phase velocity and attenuation. Int. J. Solids Struct. 41, 4587–4597. Sharma, M.D., 2010. Piezoelectric effects on the velocities of waves in an anisotropic piezo-poroelastic medium. Proc. R. Soc. A 466, 1977–1992. Shrout, T.R., Schulze, W., Biggers, J.V., 1979. Simplified fabrication of PZT/ polymer composites. Mater. Res. Bull. 14, 1553–1559. Vashishth, A.K., Gupta, V., 2009a. Vibrations of porous piezoelectric ceramic plates. J. Sound Vib. 325, 781–797. Vashishth, A.K., Gupta, V., 2009b. Wave propagation in transversely isotropic porous piezoelectric materials. Int. J. Solids Struct. 46, 3620– 3632. Vashishth, A.K., Gupta, V., 2011a. Uniqueness theorem, theorem of reciprocity, and eigenvalue problems in linear theory of porous piezoelectricity. Appl. Math. Mech. Eng. Ed. 32, 479–494. Vashishth, A.K., Gupta, V., 2011b. Reflection and transmission of plane waves from a fluid-porous piezoelectric solid interface. J. Acoust. Soc. Am. 129, 3690–3701. Vashishth, A.K., Gupta, V., 2012. Scattering of ultrasonic waves from porous piezoelectric multilayered structures immersed in a fluid. Smart Mater. Struct., 1–18, 125002. Vashishth, A.K., Gupta, V., 2013. Ultrasonic wave’s interaction at fluidporous piezoelectric layered interface. Ultrasonics 53, 479–494. Wang, Q., Chen, X., Zhu, J., Darvell, B.W., Chen, Z., 2008. Porous Li-Na-K niobate bone-substitute ceramics: microstructure and piezoelectric properties. Mater. Lett. 62, 3506–3508. Xia, Z., Ma, S., Qiu, X., Wu, Y., Wang, F., 2003. Influence of porosity on the stability of change and piezoelectricity for porous polytetrafluoroethylene film electrets. J. Electrost. 59, 57–69. Zeng, T., Dong, X.L., Mao, C.L., Zhou, Z.Y., Yang, H., 2007. Effects of pore shape and porosity on the properties of porous PZT 95/5 ceramics. J. Eur. Ceram. Soc. 27, 2025–2029.
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Mechanics of Materials journal homepage: www.elsevier.com/locate/mechmat
3-D waves in porous piezoelectric materials Anil K. Vashishth ⇑, Vishakha Gupta Department of Mathematics, Kurukshetra University, Kurukshetra 136 119, India
a r t i c l e
i n f o
Article history: Received 1 May 2014 Received in revised form 14 August 2014 Available online 2 October 2014 Keywords: Anisotropic Phase velocity Porosity Porous piezoelectric materials Wavefront Waves
a b s t r a c t Wave propagation in porous piezoelectric materials, possessing crystal symmetries monoclinic (2, m), orthorhombic (222, 2 mm), tetragonal (4), trigonal (32), hexagonal mÞ, is studied. The Christoffel equation is derived for 3D waves in (6 mm) and cubic (43 an anisotropic porous piezoelectric medium. The four roots of the biquadratic equation give the complex wave velocities of four waves propagating in such a medium. These complex wave velocities are resolved to obtain the phase velocities and attenuation factors of waves. The algebraic implicit expressions are derived for monoclinic (2, m), orthorhom m) crystal bic (222, 2 mm), tetragonal (4), trigonal (32), hexagonal (6 mm) and cubic (43 classes. The characteristics of waves in porous piezoelectric materials are studied in terms of the velocity surfaces and attenuation surfaces. The effects of phase direction, frequency, piezoelectricity, porosity and crystal symmetry on the velocity surfaces and attenuation surfaces are investigated for these crystal classes. The effects of phase direction and crystal symmetry on the skewing angles and wave fronts of quasi waves are also studied. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Piezoelectric materials possess the important property of linear coupling between mechanical and electrical fields, which renders them useful as transducers, actuators, sensors and filters, etc. in wide range engineering applications in smart structures and devices. Wave propagation in piezoelectric media has been extensively investigated in connection with the generation and transmission of disturbances in electro-acoustic devices such as transducers and resonators. Some of the notable texts on wave propagation in piezoelectric materials are Auld (1973, 1990); Nayfeh (1995) and Royer and Dieulesaint (1996). A detailed survey of wave propagation in piezoelectric materials belonging to different crystal classes is presented in the texts (Auld, 1973, 1990; Royer and Dieulesaint, 1996). Three waves
⇑ Corresponding author. E-mail addresses: [email protected] [email protected] (V. Gupta). http://dx.doi.org/10.1016/j.mechmat.2014.09.002 0167-6636/Ó 2014 Elsevier Ltd. All rights reserved.
(A.K.
Vashishth),
viz. quasi P, quasi S1 and quasi S2 waves propagate in anisotropic piezoelectric materials. One of the essential features of the wave phenomenon in such materials is the stiffened waves which exist as a result of electromechanical coupling. The phase velocities and slowness do depend upon directions of propagation, crystal symmetry, poling directions and other features of the medium. Kyame (1949) studied the wave propagation in piezoelectric materials by taking quasi-static electric field approximation into account. The propagation of waves at large distances from a source of disturbance in an infinite piezoelectric medium of hexagonal symmetry was investigated by Rao (1978). Auld (1981) presented a short survey related to wave propagation and resonance phenomena in piezoelectric materials and relates the concepts and theory to the physical properties and crystal symmetry of the materials. Special features peculiar to wave propagation in piezoelectric materials were noted and a brief sketch of the methods used for solving piezoelectric boundary value problems was also given. It was found that an extra
A.K. Vashishth, V. Gupta / Mechanics of Materials 80 (2015) 96–112
solution appears due to piezoelectric stiffening terms when only one of the components of the wave vector is specified. Christoffel equations for electroacoustic waves in unbounded piezoelectric crystals were solved by Every (1987) for the weak and strong electromechanical coupling cases. In general, waves in piezoelectric materials are quasi waves. However, for some types of material symmetry, pure longitudinal or pure transverse wave modes exist for certain specified directions of propagations (Romeo, 1996). Daros and Antes (2000) derived the strong ellipticity conditions for piezoelectric materials of the crystal symmetry classes 6 mm and 622 using positivity conditions for quadratic, cubic and quartic polynomials. The differential equation for the wavefront shape due to a line source in cylindrically hexagonal piezoelectric solids was obtained by Daros (2000). In spite of the great advances in the development of single phase piezoelectric materials with high hydrostatic sensitivity, such materials have shortcomings such as large hydrostatic piezoelectric coefficients and flexibility, high hydrostatic sensitivity and low acoustic impedance, etc. However, due to high hydrostatic figure of merit and low acoustic impedance, porous piezoelectric materials have been of great interest and technological importance in ultrasonic applications such as hydrophones, actuators, miniature accelerometer and underwater transducers (Shrout et al., 1979; Arai et al., 1991). Use of the piezoelectric effect in porous piezoelectric ceramics offers an original method for studying the coupling between the electrical, mechanical, permeability and of course piezoelectric properties of porous systems. Different analytical models (Matsunaka et al., 1988; Banno, 1993; Gomez and Montero, 1996a) have been developed to study the effects of pore volume fraction and the connectivity on the elastic, dielectric and piezoelectric properties of porous piezoelectric materials. Experimental studies (Roncari et al., 2001; Xia et al., 2003; Boumchedda et al., 2007; Levassort et al., 2007; Zeng et al., 2007; Lee et al., 2008; Wang et al., 2008; Boumchedda et al., 2010) have been done related to characteristics, fabrication and manufacturing of porous piezoelectric materials and the influence of porosity on its properties. The study of the effects of porosity on the electromechanical properties of porous piezoelectric materials, on the basis of numerical models based on finite-element method, was done by Kar-Gupta and Venkatesh (2006, 2007, 2013). Lacour et al. (1994) presented a theory to study the effects of compressibility and permeability of the viscous fluid saturating the porous piezoelectric materials on the piezoelectric properties of piezoelectric ceramics. Gomez and Montero (1996b, 1997a,b) dealt with a new system of constitutive relations that describe the elastic, dielectric and the piezoelectric behavior of porous piezoelectric materials. The influence of the coupling mechanisms on the material parameters was also analyzed. Craciun et al. (1998) made an experimental study on wave propagation in porous piezoelectric ceramics in order to improve their properties over dense ceramics. The effects of porosity on phase velocity and attenuation of longitudinal waves were
97
also studied experimentally. Gomez et al. (2000) adopted a 2D model to study the propagation of acoustic plane waves through composite materials using the finite-element method. Altay and Dokmeci (2005) expressed the governing equations of a porous piezoelectric continuum in variational form and obtained a three field variational principle with some conditions. A survey of the literature related to porous piezoelectric materials reveals that very few authors have established theoretical models for porous piezoelectric materials. Vashishth and Gupta (2009a) derived the constitutive equations and equations of motion for porous piezoelectric materials. In order to improve the properties of porous piezoelectric materials, the knowledge of their physical properties and wave phenomena in such a medium is desirable. An analytical study of wave propagation in transversely isotropic porous piezoelectric materials was done by them (Vashishth and Gupta, 2009b). The effects of porosity, direction of propagation and piezoelectricity on the phase velocities, slowness and attenuation coefficients were investigated therein. Sharma (2010) analyzed the effects of piezoelectricity on the phase velocities and group velocities of waves propagating in an anisotropic porous piezoelectric material. Subsequently, the uniqueness theorem, theorem of reciprocity and general theorems in the linear theory of porous piezoelectricity were established (Vashishth and Gupta, 2011a). Based on the theoretical formulation developed, Vashishth and Gupta, 2011b, 2012, 2013 carried out studies of reflection and transmission of waves in models involving porous piezoelectric materials. Three-dimensional wave propagation in porous piezoelectric materials, for different symmetry classes, is studied in this paper. The porous piezoelectric material is assumed to be saturated with a viscous fluid. The Christoffel equation, corresponding to an anisotropic porous piezoelectric medium, is derived. The wave velocities of four inhomogeneous waves are obtained. These wave velocities give further the phase velocities and attenuation coefficients of the corresponding waves. The results for different crystal classes, viz. monoclinic (2, m), orthorhombic (222, 2 mm), tetragonal (4), trigonal (32), hexagonal (6 mm) and cubic (43mÞ are deduced. The effects of phase direction, frequency, piezoelectricity, porosity and crystal symmetry on the velocity surfaces and attenuation surfaces are investigated. The effects of phase direction and crystal symmetry on the skewing angles and wave fronts are also studied. 2. Christoffel equation for anisotropic porous piezoelectric materials A porous piezoelectric material saturated with a viscous fluid is considered. The constitutive equations (Vashishth and Gupta, 2009a) for anisotropic porous piezoelectric materials are
r ¼ c:e þ me e:E f:E ;
ð1Þ
r ¼ m:e þ Re ~f:E e :E ;
ð2Þ
98
A.K. Vashishth, V. Gupta / Mechanics of Materials 80 (2015) 96–112
D ¼ e:e þ ~fe þ n:E þ A:E ;
ð3Þ
D ¼ f:e þ e e þ A:E þ n :E ;
ð4Þ
where rðr Þ and eðe Þ are the stress and strain tensors of order 2 acting on solid (fluid) phase of porous aggregate. EðE Þ and DðD Þ are electric field and electric displacement vectors. c is the elastic stiffness tensor of order 4. The elastic constant R measures the pressure to be exerted on fluid to push its unit volume into the porous matrix. eðe Þ and nðn Þ are piezoelectric and dielectric tensors are of order 3(1) and 2, respectively. The tensors m; f; ~f; and A of order 2; 3,1; and 2 link the elastic; piezoelectric; and dielectric properties of two phases of the porous aggregate. According to Biot, 1956a,b theory, homogeneous porous solid matrix and its saturating fluid are treated in the manner of two interpenetrating elastic continua. Losses arise due to the viscous motion of the fluid relative to the solid matrix. The equations of motion and Gauss equations for such a material are
r:r ¼ q11 :u þ q12 :u þ b:ðu_ u_ Þ;
ð5Þ
r:r ¼ q12 :u þ q22 :u b:ðu_ u_ Þ;
ð6Þ
r:D ¼ 0;
ð7Þ
r:D ¼ 0;
ð8Þ
where uðu Þ is the mechanical displacement vector. UðU Þ is the electric potential. q11 ; q12 and q22 are dynamical tensors of order 2 related to the porous aggregate density (q), pore fluid density (qf Þ and the inertial coupling between the phases. In the low frequency range, the dissipation tensor (b) of order 2 for an anisotropic porous medium is related to porosity (f), fluid viscosity (l) and solid-matrix permeability (v) as
b¼f
2
lv1 :
ð9Þ
In case of high frequency, the friction force of the fluid on the solid becomes out of phase with relative rate of flow and exhibit’s frequency dependence. In high frequency range, l is replaced by lFðjÞ, where the complex function FðjÞ may be defined as (Badiey and Yamamoto, 1985)
FðjÞ ¼
pffiffiffiffiffiffi pffiffiffiffiffiffi ij tanh ij 1 pffiffiffiffiffiffi pffiffiffiffiffiffi ; 3 1 tanh ij = ij
ð10Þ
pffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where j ¼ 6p v0 tqf =lf ; v0 is the norm of the permeability matrix and t is the wave frequency. For plane harmonic waves, the solution of the system (5)–(8) can be considered as
ðu; u ; U; U Þ ¼ ðB; F; G; HÞ exp
n
ix
v
:x t ;
ð11Þ
where i ¼ 1. x is the circular frequency of harmonic waves and v is the wave velocity in the medium. The direction of phase propagation (n) of a wave can be written as
where h is the angle made by the direction of phase propagation with the x3 axis and / is the azimuth angle. Making use of the constitutive Eqs. (1)–(4) and the Eq. (11) in the Eqs. (5)–(8), one can obtain
11 ÞB þ ðK2 h q 12 ÞF ¼ 0; ðK1 h q
ð13Þ
12 ÞB þ ðK4 h q 22 ÞF ¼ 0; ðK3 h q
ð14Þ
where
B ¼ ½B1 B2 B3 T ; F ¼ ½F 1 F 2 F 3 T ;
0
K1 ¼ P þ e0 r0 þ f0 b ;
K2 ¼ ðm:nÞ n þ e0 t0 þ f0 z0 ; 0
K3 ¼ ðm:nÞ n þ n0 r0 þ n b ; K4 ¼ k4 n n; h ¼ 1=v2 ; Pik ¼ cijkl nl nj ; e0k ¼ eikl ni nl ; f0k ¼ fikl ni nl ; n0 ¼ ð~f:nÞn; n ¼ ðe :nÞn; n0 ¼ n:ðn nÞ; n0 ¼ n :ðn nÞ; a ¼ A:ðn nÞ; f0 n0 n0 f0 0 ; 2 0 0 e a a nn a n n0 n0 n 0 ; t0 ¼ 2 0 0 n a a nn a
r0 ¼
0
b ¼
a n0 f0 a n0 n 0 0 0 ; z ; e ¼ n a a a2 n0 n0 a2 n0 n0
k4 ¼ R þ ~f:n½e :n=a n0 ð~f:n e :nn0 =aÞ=ða2 n0 n0 Þ þ e :nað~f:n e :nn0 =aÞ=ða2 n0 n0 Þ; q q b ¼ 22 ¼ q22 þ b; 12 ¼ q12 b; q 11 ¼ q11 þ b;
i b: x
The system (13) and (14) can be written as
CB ¼ 0;
ð15Þ
where
C ¼ B0 h þ C0 þ
D0 ; c1 h þ c2
ð16Þ
X2 1 c2 11 ; C0 ¼ q X3 X2 ; c1 c1 c1 c2 c2 D0 ¼ X4 þ X3 X2 ; c1 c1 B0 ¼ K1
12 þ K2 G0 K3 q 12 F0 K3 ; X2 ¼ K2 F0 q 12 þ q 12 q 12 G0 K3 ; 12 F0 q X3 ¼ K2 G0 q
2
n ¼ ðsin h cos /; sin h sin /; cos hÞ;
ð12Þ
12 ; 12 G0 q X4 ¼ q c1 ¼ k4 n21 d1 þ k4 n22 d4 þ k4 n23 d6 þ 2n1 n2 k4 d2 þ 2n2 n3 k4 d5 þ 2n1 n3 k4 d3 ;
A.K. Vashishth, V. Gupta / Mechanics of Materials 80 (2015) 96–112 0 22 22 22 22 c2 ¼ ðq 22 d4 þ q12 d2 þ q23 d5 Þ; G ¼ adjðq Þ; 2
22 22 22 d1 ¼ q 22 q33 ðq23 Þ ;
as stiffened qS1 and qS2 waves respectively. Correspond1 1 1 ingly, Q 1 represent attenuation 1 ; Q 2 ; Q 3 and Q 4 coefficients of qP 1 ; qP 2 ; qS1 and qS2 waves respectively.
22 22 22 22 d2 ¼ q 13 q23 q12 q33 ;
22 22 22 22 d3 ¼ q 12 q23 q13 q22 ; 2
22 22 22 d4 ¼ q 11 q33 ðq13 Þ ;
3. Numerical results and discussion Numerical computation of the analytical expressions of phase velocities, attenuation coefficients and skewing angles of stiffened qP 1 ; qP 2 ; qS1 and qS2 waves was done for particular models whose data is listed in the Tables 1–6.
22 22 22 22 d5 ¼ q 12 q13 q11 q23 ;
2
22 22 22 d6 ¼ q 11 q22 ðq12 Þ ; 2 22 22 2 22 F 011 ¼ ðq 33 n2 q22 n3 þ 2n2 n3 q23 Þk4 ; 2 22 22 22 22 F 012 ¼ ðq 12 n3 þ q33 n1 n2 q13 n2 n3 q23 n1 n3 Þk4 ;
F 013 F 022
¼
3.1. Skewing angles
2 22 22 22 ð 22 13 n2 23 n1 n2 12 n2 n3 þ 22 n1 n3 Þk4 ; 2 22 2 22 ð 22 33 n1 11 n3 þ 2 13 n1 n3 Þk4 ;
¼ q
q
q
q
q q
q
2 22 22 22 22 F 023 ¼ ðq 23 n1 q13 n1 n2 þ q11 n2 n3 q12 n1 n3 Þk4 ; 2 22 22 2 22 F 033 ¼ ðq 22 n1 q11 n2 þ 2q12 n1 n2 Þk4 :
The condition of existence of non-trivial solution of the system (15) leads to 4
3
2
T 1 h þ T 2 h þ T 3 h þ T 4 h þ T 5 ¼ 0;
ð17Þ
where T 1 ; T 2 ; T 3 ; T 4 and T 5 are given in the Appendix A). The coefficients of the Eq. (17) and other implicit expressions within these coefficients for the monoclinic (2, m), orthorhombic (222, 2 mm), tetragonal (4), trigonal (32), mÞ crystal classes can be hexagonal (6 mm) and cubic (43 deduced from the corresponding expressions for anisotropic case and these are detailed in the Appendix B). The solution of the Eq. (17) gives the four complex roots hj ; ðj ¼ 1; 2; 3; 4Þ and correspondingly wave velocities vj of four waves are obtained. The phase velocities (Vj ; j ¼ 1; 2; 3; 4Þ and attenuation coefficients (Q 1 j ; j ¼ 1; 2; 3; 4Þ for these waves are defined as
Vj ¼
ðReðvj ÞÞ2 þ ðImðvj ÞÞ2 ; Reðvj Þ
Q 1 ¼ 2Imðvj Þ=Reðvj Þ: j
ð18Þ
ð19Þ
Corresponding to the each root hj ; ðj ¼ 1; 2; 3; 4Þ, the homogeneous system (15) is solved for a non-trivial ðjÞ ðjÞ ðjÞ solution ðB1 ; B2 ; B3 Þ, which defines the polarization direction of the jth wave propagating in a porous piezoelectric material. The skewing (deviation) angle of the polarization direction of jth wave from the phase propagation direction is defined as
cj ¼ cos1 ðBðjÞ :nÞ:
99
ð20Þ
Thus, for a given phase direction, four plane harmonic waves propagate in an anisotropic porous piezoelectric material. In general, the polarization vectors of these waves are neither perpendicular nor parallel to the phase direction, so these waves are termed as quasi waves. The wave with the largest phase velocity V1 is termed as stiffened qP 1 wave and the wave with the smallest phase velocity V2 is termed as stiffened qP2 wave. The other two waves, with phase velocities V3 and V4 , are named
The range of skewing angles c1 ; c2 ; c3 and c4 for all the considered crystal classes is shown in the Table 7. It is clear from the Table that, the skewing angles associated with the qP1 and qP 2 waves are not equal to 0° and that associated with qS1 and qS2 waves are not equal to 90° , so these waves are termed as quasi–longitudinal and quasi-shear waves respectively. The smallest skewing angle is associated with the fastest qP1 wave. In case of 6 mm class, qS2 wave is a pure transverse wave in all the directions of propagation which validates the present model’s results for hexagonal symmetry. In the crystal class 43m, the skewing angles associated with qP1 and qP2 waves are near to 0° and those corresponding to qS1 and qS2 are near to 90° meaning thereby that qP 1 and qP 2 are nearly longitudinal and qS1 and qS1 are nearly transverse waves. 3.2. Phase velocities The phase velocities of qP 1 ; qP 2 ; qS1 and qS2 waves were computed in low frequency range (LFR) and high frequency range (HFR). The value of t is considered as 50 Hz in LFR and 1 MHz in HFR. Fig. 1(a) and (b) exhibit the variations of phase velocities of the four waves in porous piezoelectric materials, saturated with viscous fluid, for the crystal classes m and 2, respectively with phase direction. The elastic, dielectric and dynamical constants of these two classes are same but piezoelectric constants are different. Elevations and depressions of phase velocity surfaces from the horizontal plane in any plot represent the velocity anisotropy for the corresponding wave in the medium. The effects of azimuth variations on quasi shear waves are more prominent in comparison to the quasi longitudinal waves. The phase velocity of qP 1 wave increases with h but the phase velocity of qP 2 wave decreases with an increase in h . The phase velocity of qS1 wave first decreases with an increase in h and attains a local minimum at h ¼ 30 but, after that, it increases with h. Contrary to this, V4 increases first and then decreases having a minimum at 45° and then increases further. The difference in the velocities of qS1 and qS2 waves reveals the anisotropy effect of monoclinic symmetry. Comparison of the respective graphs in Fig. 1(a) and (b) reveals that the phase velocities of qP 1 ; qS1 and qS2 waves in both the classes (m and 2) of monoclinic symmetry are not significantly different. The variation pattern of phase velocity of qP 2 wave also remains same but its magnitude for class m is less than that for class 2. However, such a phenomenon
100 Table 1 Density q
A.K. Vashishth, V. Gupta / Mechanics of Materials 80 (2015) 96–112
ðKg=m3 Þ, elastic stiffness coefficients cIJ ðGPaÞ and elastic coupling coefficients mij ðGPaÞ for different crystal classes (Auld, 1973).
Crystal Class
Material
q
c11
c12
c13
c16
c22
c23
c26
c33
c36
c44
c45
c55
c66
m11
m12
m22
m33
2, m
Barium Sodium Niobate Rochelle Salt Barium Sodium Niobate Barium Titanate Barium Titanate Bismuth Germanate
5300
239
104
50
4.2
247
52
5.7
135
3.2
65
5.4
66
76
8.8
11.5
16.8
5.2
1767
28
17.4
15
0
41.4
19.7
0
39.4
0
6.66
0
2.85
9.6
8.8
0
16.8
5.2
5300
239
104
50
0
247
52
0
135
0
65
0
66
76
8.8
0
16.8
5.2
6020
275
179
151
50
275
151
50
165
0
54.3
0
54.3
113
8.8
0
8.8
5.2
5700
150
66
66
0
150
66
0
146
0
44
0
44
42
8.8
0
8.8
5.2
7095
115.8
27
27
0
115.8
27
0
115.8
0
43.6
0
43.6
43.6
8.8
0
8.8
8.8
222 2 mm
4 6 mm m 43
Table 2 Density q
ðKg=m3 Þ, elastic stiffness coefficients cIJ ðGPaÞ and elastic coupling coefficients mij ðGPaÞ for crystal class 32 (Auld, 1973).
Crystal Class
Material
q
c11
c12
c13
c14
c15
c24
c25
c33
c44
m11
m33
32
Quartz
2651
86.74
6.99
11.91
17.91
0
17.91
0
107.2
57.94
8.8
5.2
Table 3 Piezoelectric constants eiJ , ei and ~fi for different crystal classes; units = C=m2 (Auld, 1973). Crystal Class e14
e15
e24
e25
e31
e32
e33
e36
e3
f14
f15
f24
f25
f31
f32
f33
f36
~f3
2 222 2 mm 4 6 mm m 43
2.8 0 2.8 21.3 11.4 0
3.4 0 3.4 21.3 11.4 0
4.2 0.154 0 15 0 0.038
0.4 0 0.4 2.74 4.35 0
0.3 0 0.3 2.74 4.35 0
4.3 0 4.3 3.7 17.5 0
5.6 0.115 0 0 0 0.038
0.6 0 0.6 0.6 0.6 0
0.12 0.06 0 0.375 0 .001
0.07 0 0 0.53 0.285 0
0.09 0 0.085 0.53 0.285 0
0.11 .0004 0 0.38 0 0.001
0.01 0 0.01 0.07 0.11 0
0.008 0 0.008 0.069 0.11 0
0.108 0 0.108 0.093 0.093 0
0.14 0.003 0.003 0 0 0.001
0.75 0 0.75 0.75 0.75 0
4.8 2.23 0 15 0 0.038
Table 4 Piezoelectric constants eiJ , ei and ~fi for the crystal classes m and 32; units = C=m2 (Auld, 1973). Crystal Class
Piezoelectric constants
m
e11 e23 f11 f23
32
e11 ¼ 0:171
¼ 0:2 ¼ 0:46 ¼ 0:005 ¼ 0:012
e12 e26 f12 f26
¼ 0:07 ¼ 0:89 ¼ 0:0018 ¼ 0:022
e13 e34 f13 f34
e14 ¼ 0:0436
¼ 0:32 ¼ 0:12 ¼ 0:008 ¼ 0:003
e3 ¼ 0:6
¼ 0:067 ¼ 0:67 ¼ 0:0017 ¼ 0:017
f11 ¼ 0:004
Table 5 Dielectric constants nij ðnC=VmÞ; nij ðnC=VmÞ and Aij ðnC=VmÞ for different crystal classes,
2, m 222 2 mm 4 32 6 mm m 43
e16 e35 f16 f35
e21 ¼ 0:4 e1 ¼ 2:6 f21 ¼ 0:01 ~f1 ¼ 1:2
e22 ¼ 0:23 e2 ¼ 1:7 f22 ¼ 0:0058 ~f2 ¼ 0:4
f14 ¼ 0:0011
~f3 ¼ 0:4
e0 ¼ 8:854 1012 is the absolute permittivity (Auld, 1973).
n11 =e0
n12 =e0
n22 =e0
n33 =e0
n11
n12
n22
n33
A11
A12
A21
A22
A33
1220 205 2.35 2920 4.52 1450 16
1107 0 0 0 0 0 0
1672 9.6 247 2920 4.52 1450 16
1480 9.5 51 168 4.68 1700 16
0:038 0:038 0:038 0.038 0.038 0.038 0.038
0:09 0 0 0 0 0 0
0:055 0:055 0:055 0.038 0.038 0.038 0.038
0:049 0:049 0:049 0.049 0.049 0.049 0.038
0:018 0:018 0:018 0.018 0.018 0.018 0.018
0:04 0 0 0.04 0 0 0
0:08 0 0 0.04 0 0 0
0:031 0:031 0:031 0.018 0.018 0.018 0.018
0:015 0:015 0:015 0.015 0.015 0.015 0.018
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v11 ¼ 1:0 10 q11 =q ¼ 0:66
qf ¼ 1000Kg=m3 10
m
10
2
v12 ¼ 0:5 10 q12 =q ¼ 0:15
l ¼ 1 103 Ns=m2 m
10
2
v22 ¼ 1:3 10 q22 =q ¼ 0:64
m
f ¼ 0:2 2
v33 ¼ 0:8 1010 m2 –
Table 7 Range of skewing angle of all the waves in different crystal classes. Crystal class
m 2 222 2 mm 4 32 6 mm m 43
Range of skewing angle qP 1 wave
qP 2 wave
qS1 wave
qS2 wave
0:8 19 0:8 19 0:9 15 0:7 15:3 0:2 16 0:06 21:4 0:002 2:1 0:01 0:32
0:3 14 0:6 30 13 29 1:1 39 9:1 25 0:3 38 6:8 22 0:04 0:97
74:2 89 74:4 89 71 89 79:3 89 74:8 89 69:5 89 87:5 89 89:6 89:9
74 89 74:2 89 77 89 78 89 75:5 89 71:7 89 90 89:8 89:9
does not exist in a poro-elastic medium. The velocities of all the quasi waves increase as the frequency shifts from LFR to HFR. The pattern of variation of phase velocities with the phase directions remains unaffected in either case. Phase velocities variations with directions and with frequency, for the crystal classes 222, 2 mm, 4, 32, 6 mm and m, are shown in Fig. 1(a) and (h). It is observed that the 43 velocities of all the waves increase as the frequency shifts from LFR to HFR in all the crystal classes except the crystal classes 2 mm and 4. The noticed changes in the phase velocities are due to changes in crystal classes and crystal type both. It is interesting to note that the azimuthal variations do considerable affect the phase velocities of all waves in the case of crystal classes 222, 4 and 32 Fig. 1(c), (e) and (f). The azimuthal variation has no effects on velocity pattern in case of 6 mm class, which confirms the results (Fig. 1(g)). Unlike the above mentioned crystal classes, velocity surfaces for LFR and HFR in case of cubic m are found exactly parallel. The effects of polar class 43 angle on the phase velocities of all the waves are significant in all the crystal classes. To study the effects of anisotropy on the variation of phase velocities, the results are deduced numerically for m from the crystal classes 222, 2 mm, 4, 6 mm and 43 the Barium Sodium Niobate crystal of class 2 but the plots m to are shown here only for the classes 222, 4 and 43 avoid the repetition. The magnitude as well as the variation pattern of qP1 wave remains unaffected in all the crystal classes. The velocity of qP 2 wave changes with crystal classes. Noticeable deviation is observed from class 2 to 222 (Fig. 2(a)). Exactly similar variation of V 2 is observed for 2 to 43mclass and so is not shown here. However, the velocity of qP2 wave is found to be same in case of 2, 2 mm, 4 and 6 mm classes. The phase velocities of qP1 ; qS1 and qS2 waves are same for 222 and 2 mm classes. qS1 velocity variations with directions is different for 2 and 222 crystal classes (Fig. 2(a)). The effects of crystal
symmetry on the phase velocity of qS1 wave are more prominently observed in cases of the crystal classes 4 and 43m (Fig. 2(b) and (c)). It is observed that velocity surfaces of all waves coincide for 2 and 6 mm classes. In m, the all the crystal classes, except the crystal class 43 magnitude and variation pattern of phase velocity of qS2 wave also remains same except for small values of polar angle. Thus, it may be concluded that the effects of crystal symmetry on the magnitudes of phase velocities are least in case of fastest waves. Fig. 3 shows the effects of pore volume fraction on the phase velocities of quasi waves in LFR for the crystal class m. The effects of pore volume fraction on the phase velocities are not significant in the HFR. It is observed that in case of LFR, the phase velocities of all the waves decrease with the increase of porosity in all the phase directions. Percolation theory (Craciun et al., 1998) may be able to explain the decrease in phase velocity with an increase of porosity. In this theory, a geometric phase transition occurs in the percolation system when the medium becomes disconnected due to the progressive dilution. The phase velocities of qP 1 ; qS1 and qS2 waves in the LFR do not change with the increase of porosity beyond 0.6 approximately. However, the slowest wave is found to be more sensitive to changes in the porosity of the medium. Thus, it can be concluded that introduction of pores in piezoceramics affects the wave velocities but only up to a certain level of porosity. Similar velocity patterns were also observed for other crystal classes and results for those classes have been found not significantly different and thus are not shown here. The wavefronts (slowness surfaces) were obtained for all different crystal classes but these are shown for the crystal classes m, 222, 4 and 32 in Fig. 4(a) and (d). The acoustic wavefronts play a central role in the analysis of many crystal acoustic phenomena, including phonon focusing, reflection and transmission at surfaces, and surface and interface waves [8]. Any of the two wavefronts of the four
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Fig. 1. Effects of frequency on the variation of phase velocities of quasi waves with phase direction; (a) Barium Sodium Niobate crystal of class m, (b) Barium Sodium Niobate crystal of class 2, (c) Rochelle Salt crystal of class 222, (d) Barium Sodium Niobate crystal of class 2 mm, (e) Barium Titanate crystal m. of class 4, (f) Quartz crystal of class 32, (g) Barium Titanate crystal of class 6 mm, (h) Bismuth Germanate crystal of class 43
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Fig. 1 (continued)
103
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Fig. 2. Anisotropic variation of phase velocities of quasi waves for Barium Sodium Niobate crystal: Reduced from the crystal class 2; (a) Class 222, (b) Class 4 m. (c) Class 43
Fig. 3. Effects of porosity on the anisotropic variation of phase velocities of quasi waves for Barium Sodium Niobate crystal of class m.
waves may meet in a number of isolated directions. There are three types of singularities (Crampin, 1981): kiss singularities, where the two sheets touch tangentially with either convex or concave contact; intersection singularities,
where the two sheets may be considered as cutting each other along a closed curve; and point singularities, where the two sheets have common vertices of cone-shaped projection of surfaces. The number of singularities as well
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105
Fig. 4. Wavefront of quasi waves; (a) Barium sodium Niobate crystal of class m, (b) Rochelle salt crystal of class 222, (c) Barium Titanate crystal of class 4, (d) Quartz crystal of class 32.
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Table 8 Two-shear wavefronts singularities directions in different crystal classes. Crystal Class
Two-shear wavefronts singularities directions
m 2 2 mm 4
ð29 ; 67 Þ; ð30 ; 66 Þ & ð30 ; 67 Þ ð29 ; 65 Þ; ð29 ; 66 Þ & ð29 ; 67 Þ h ¼ 29 ; 84 6 / 6 90 ; h ¼ 30 ; 84 6 / 6 90 h ¼ 0 ; 0 6 / 6 90 ; h ¼ 1 ; 0 6 / 6 90 ; h ¼ 2 ; 34 6 / 6 52 h ¼ 0 ; 0 6 / 6 90 & ð67 ; 30 Þ 0 6 h 6 20 ; 0 6 / 6 90
32 6 mm
is summarized in the Table 8. In case of the crystal classes 222 and 4, the wavefronts of qP2 and qS2 waves also have common points of contact. In case of the crystal class 222, qP2 and qS2 waves have common phase velocities in the direction ð68 ; 80 Þ while in case of the crystal class 4, these two waves have common phase velocities in the directions h ¼ 35 ; / ¼ 74 76 ; h ¼ 36 ; / ¼ 70 72 and h ¼ 37 ; / ¼ 68 69 . 3.3. Attenuation coefficients
as their types may vary due to changes in crystal symmetry. qS1 and qS2 waves have common phase velocities in different phase directions in different crystal classes, which
The attenuations of qP1 ; qP 2 ; qS1 and qS2 waves in PPM saturated with viscous fluid are computed for different crystal classes in the LFR and HFR. In the LFR, the attenuation is
Fig. 5. Variation of attenuation coefficients with the pore volume fraction for Barium Sodium Niobate crystal of class m; (i) LFR (ii) HFR.
Fig. 6. Variation of phase velocities of quasi waves with phase direction; porous non-piezoelectric materials.
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Fig. 7. Effects of piezoelectricity on the variation of phase velocities of quasi waves with phase direction; (a) Barium Sodium Niobate crystal of class 2, (b) Rochelle salt crystal of class 222, (c) Barium Sodium Niobate crystal of class 2 mm, (d) Barium Titanate crystal of class 4, (e) Barium Titanate crystal of class 6 mm.
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Fig. 7 (continued)
mainly contributed by viscosity losses due to the friction at the fluid-grain boundary. The attenuation of the slowest wave is maximum while that of the fastest wave is minimum. It is observed that the variation pattern of attenuations of qP1 and qP2 waves with h is same as for phase velocities while attenuations of qS1 and qS2 waves behave opposite to phase velocities variation and thus plots are omitted here. It is also observed that unlike the phase velocities, the attenuations of all the waves decrease as the frequency shifts from LFR to HFR. The effects of azimuth variation on the attenuation coefficient of qP1 wave are more significant for high value of h while these effects are found significant for small values of h in case of qS1 and qS2 waves. The effects of electro-elastic interaction on the attenuation coefficients in the HFR for different crystal classes are studied. It is found that the electro-elastic interaction has very small effects on the magnitude of attenuation coefficients and pattern of variation thereof in all the crystal classes except the crystal classes 4 and 6 mm. In case of classes 4 and 6 mm, the attenuations of qP 1 ; qS1 and qS2 waves increase due to inclusion of piezoelectric effect like the phase velocities while that of qP 2 wave decreases due to these effects unlike the phase velocity and thus the plots are not shown here. Except qP1 wave, the electro-elastic interaction has significant effects on the pattern of variation of attenuation coefficients of all the waves with the phase direction. Thus, it can be inferred that the effects of electro-elastic interaction on the attenuation of waves are different in different materials having different kinds of symmetry. Fig. 5(i) and (ii) show the dependence of the attenuation coefficients on the pore volume fraction for the crystal class m in the LFR and HFR, respectively. The value of h and / are considered as 30 and 60 , respectively. It is observed that in the LFR, the attenuation of qP 1 ; qS1 and qS2 waves first increase with porosity and attain local maxima at 35% porosity and after that it decreases monotonically with an increase in porosity. However, the attenuation of qP2 wave increases monotonically with porosity and its attenuation is the maximum of all the waves in LFR, which is so. However, in the HFR, the attenuation of qP2 wave is comparable to the other waves, which is in conformity with the fact that in the high frequency range
qP2 wave is propagating in nature. The attenuation of all the waves increases rapidly with porosity in the HFR. Similar results were also observed for other crystal classes and results for those classes have been found not significantly different and thus are not shown here. 4. Particular cases In order to validate the considered model of study here, the earlier established results are obtained. as particular cases of the present model, which are described as follows: 4.1. 3-D waves in porous non-piezoelectric materials The correspondence between porous piezoelectric materials and porous non-piezoelectric materials can be obtained by substitution of e; f; e ; ~f; n; n ; A; D; D ; E; E ; G; H as zero. The homogeneous system for porous nonpiezoelectric materials can be obtained from the homogeneous system (15), derived for porous piezoelectric materials, by substitution of e; f; e ; ~f; n; n ; A; D; D ; E; E ; G; H as zero. The condition of existence of non-trivial solution of the homogeneous system, corresponding to porous non-piezoelectric material case, also leads to biquadratic equation in h. Thus, the number of waves remains unaffected due to the effects of piezoelectricity. The phase velocities of all the quasi waves are obtained for porous non-piezoelectric materials by neglecting effects of piezoelectricity and considering the data for elastic constants and other parameters listed in Sharma (2004). The variation of phase velocities of waves propagating in porous non-piezoelectric materials with the direction of propagation is shown in Fig. 6 and the results are found to be in agreement with the results of Sharma (2004). Further, to study the effects of piezoelectricity on the magnitude of phase velocities of quasi waves, the results are computed for porous non-piezoelectric materials (PM) and porous piezoelectric materials (PPM) cases. A magnificent effects of piezoelectricity are not observed in m and thus are the case of crystal classes m, 32 and 43 not shown here. The effects of piezoelectricity on the variation of phase velocities with the phase directions for
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Fig. 8. Variation of phase velocities of quasi waves with phase direction; non-porous piezoelectric material; Barium Sodium Niobate crystal of class m.
Fig. 9. Slowness surfaces cut by the x1 x2 plane; non-porous piezoelectric materials; Lithium Niobate crystal of class 3 m.
the crystal classes 2, 222, 2 mm, 4 and 6 mm are shown in Fig. 7(a) and (e). For the classes 2 and 4, electro-elastic interaction has weak effects on qP 1 ; qS1 and qS2 waves but phase velocity of qP 2 wave increases due to piezoelectricity (Fig. 7(a) and (d)). The magnitudes of phase velocities of all the waves remain almost unaffected due to piezoelectricity while the variation pattern of qS2 wave gets effected due to electro-elastic interaction in the crystal class 222 (Fig. 7(b)). In case of class 2 mm (Fig. 7(c)), the phase velocities of qP 1 ; qP 2 and qS2 waves increase due to piezoelectricity while that of qS1 wave remains unaffected. The velocity of qP1 wave increases due to piezoelectricity only for small values of polar angle while that of qS2 wave increases due to piezoelectricity except for small values of polar angle. The velocity of the slowest wave increases for all values of polar angle. Fig. 7(e) exhibits that the phase velocities of all the quasi waves increase due to electroelastic interaction in case of 6 mm class. The effects of piezoelectricity on the fastest wave are more significant in this case. Thus, it may be concluded that the different crystals having a different piezoelectric coupling have different wave velocities and consideration of piezoelectricity in the wave phenomenon is important for some of the crystal classes. It is also justified from here that Barium Titanate (6 mm) has a strong piezoelectric coupling in comparison to Quartz crystal (32). These observations are found in agreement with the experimental ones (Jaffe et al., 1971).
4.2. 3-D waves in non-porous piezoelectric materials To study the qualitative difference between 3-D waves in non-porous piezoelectric materials and porous piezoelectric materials, the homogeneous system (15), derived for porous piezoelectric materials, is reduced for non-porous piezoelectric materials by substitution of m; R; f; e ; ~f; n ; A; q12 ; q22 ; b; F; H; r ; e ; u ; E ; D as zero which further leads to a cubic equation in h unlike the case of porous piezoelectric materials. Thus, the number of waves propagating in non-porous piezoelectric materials is three unlike the PPM case. It is also found that the waves propagating in non-porous piezoelectric material are mutually orthogonal but the waves are not mutually orthogonal in case of porous piezoelectric materials. The variation of phase velocities of waves, propagating in non-porous piezoelectric materials having crystal symmetry monoclinic (m), with direction of propagation is shown in Fig. 8. The results for non-porous piezoelectric materials having crystal symmetry trigonal (3 m) have been obtained as a particular case so that present results are validated (Fig. 9). The elastic, piezoelectric and dielectric constants listed in the text Auld (1973) are considered. The variation of slowness with the direction of propagation in the x1 x2 plane is found to be in agreement with the results of Auld (1973), as should be.
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5. Conclusion The three-dimensional wave propagation in porous piezoelectric materials saturated with viscous fluid is studied. The Christoffel equation is derived for anisotropic porous piezoelectric materials and the algebraic expressions of the Christoffel coefficients are deduced for crystal m. The effects classes 2, m, 222, 2 mm, 4, 32, 6 mm and 43 of frequency, porosity, piezoelectricity and crystal symmetry on the velocity surfaces and attenuation surfaces of four waves propagating in a porous piezoelectric medium are studied numerically for particular models. The skewing angles associated with the quasi longitudinal waves are smaller in comparison to the angles associated with the quasi-shear waves. In case of 6 mm class, qS2 wave is a pure transverse wave in all the directions of propagation. The effects of polar angle and azimuth angle on the phase velocities of waves are different for different crystal classes. The effects of azimuth variations on quasi shear waves are more prominent in comparison to quasi longitudinal waves. The velocity of the slowest wave for the class m is 16% less than in comparison to the class 2 while velocities of other three waves remain same in both the classes of a monoclinic group. The azimuth variation has no effects on the phase velocities in the case of the crystal class 6 mm. The effects of crystal symmetry on the magnitudes of phase velocities are least in case of fastest waves while these effects on the other three waves are different for different crystal classes. The velocities of all quasi waves increase while attenuations decrease as the frequency shifts from LFR to HFR. The phase velocities of all waves decrease with an increase in porosity in LFR. The slowest wave is found to be more sensitive to the porosity of the medium. The introduction of pores in piezoceramics affects the wave velocities but only up to a certain level. The slowest wave is highly attenuated while the fastest wave is least attenuated. The changes in the attenuation of waves are less significant due to change of crystal symmetry. The attenuation of all the waves increases with porosity in HFR. Different crystals having a different piezoelectric coupling have different wave velocities and attenuations and consideration of piezoelectricity for wave phenomenon is important for some of the crystal classes. Barium Titanate has a strong piezoelectric coupling in comparison to Quartz crystal. The wavefronts of two quasi-shear waves and slowest quasi-shear and slowest quasi-longitudinal waves may coincide along some phase directions, and these directions get affected due to crystal symmetry. The results are obtained for the particular cases porous non-piezoelectric materials and non-porous piezoelectric materials and are found in agreement with the earlier established results. Appendix A
T 01 ¼ B011 y1 þ B012 y6 þ B013 y11 ; T 02 ¼ C 011 y1 þ C 012 y6 þ C 013 y11 þ B011 y2 þ B012 y7 þ B013 y12 ;
T 03 ¼ C 011 y2 þ C 012 y7 þ C 013 y12 þ B011 y4 þ B012 y9 þ B013 y14 ; T 04 ¼ C 011 y4 þ C 012 y9 þ C 013 y14 ;
T 05 ¼ D011 y1 þ D012 y6 þ D013 y11 þ B011 y3 þ B012 y8 þ B013 y13 ;
T 06 ¼ D011 y2 þ D012 y7 þ D013 y12 þ C 011 y3 þ C 012 y8 þ C 013 y13 þ B011 y5 þ B012 y10 þ B013 y15 ;
T 07 ¼ D011 y4 þ D012 y9 þ D013 y14 þ C 011 y5 þ C 012 y10 þ C 013 y15 ;
y1 ¼ B022 B033 B023 B032 ; y2 ¼ B022 C 033 þ C 022 B033 B023 C 032 C 023 B032 ;
y3 ¼ B022 D033 þ D022 B033 B023 D032 D023 B032 ; y4 ¼ C 022 C 033 C 023 C 032 ;
y5 ¼ C 022 D033 þ D022 C 033 C 023 D032 D023 C 032 ; y6 ¼ B031 B023 B021 B033 ;
y7 ¼ B031 C 023 þ C 031 B023 B021 C 033 C 021 B033 ; y8 ¼ B021 D033 D021 B033 þ B023 D031 þ D023 B031 ;
y9 ¼ C 021 C 033 þ C 023 C 031 ; y10 ¼ C 021 D033 D021 C 033 þ C 023 D031 þ D023 C 031 ;
y11 ¼ B021 B032 B031 B022 ; y12 ¼ B021 C 032 þ C 021 B032 B031 C 022 C 031 B022 ;
y13 ¼ B021 D032 þ D021 B032 B031 D022 D031 B022 ; y14 ¼ C 021 C 032 C 022 C 031 ;
T 1 ¼ c1 T 01 ; T 2 ¼ c2 T 01 þ c1 T 02 ; T 3 ¼ c2 T 02 þ c1 T 03 þ T 05 ; T 4 ¼ c2 T 03 þ c1 T 04 þ T 06 ; T 5 ¼ c2 T 04 þ T 07
y15 ¼ C 021 D032 þ D021 C 032 C 022 D031 D022 C 031 :
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Appendix B 2
e01 ¼ ðe15 þ e31 Þn1 n3 þ ðe25 þ e36 Þn2 n3 ; e02 ¼ ðe14 þ e36 Þn1 n3 þ ðe24 þ e32 Þn2 n3 ; e03 ¼ e15 n21 þ e24 n22 þ e33 n23 þ ðe14 þ e25 Þn1 n2 ; nk ¼ e3 n3 nk :
m
e01 ¼ e11 n21 þ e26 n22 þ e35 n23 þ ðe16 þ e21 Þn1 n2 ; e02 ¼ e16 n21 þ e22 n22 þ e34 n23 þ ðe12 þ e26 Þn1 n2 ; e03 ¼ ðe13 þ e35 Þn1 n3 þ ðe23 þ e34 Þn2 n3 ; nk ¼ ðe1 n1 þ e2 n2 Þnk .
2, m
P 11 ¼ c11 n21 þ 2c16 n1 n2 þ c66 n22 þ c55 n23 ; P12 ¼ c16 n21 þ ðc12 þ c66 Þn1 n2 þ c26 n22 þ c45 n23 ; P 13 ¼ ðc13 þ c55 Þn1 n3 þ ðc36 þ c45 Þn2 n3 ; P22 ¼ c66 n21 þ 2c26 n1 n2 þ c22 n22 þ c44 n23 ; P 23 ¼ ðc36 þ c45 Þn1 n3 þ ðc23 þ c44 Þn2 n3 ; P33 ¼ c55 n21 þ 2c45 n1 n2 þ c44 n22 þ c33 n23 ; n0 ¼ n11 n21 þ n22 n22 þ n33 n23 þ 2n12 n1 n2 ; n0 ¼ n11 n21 þ n22 n22 þ n33 n23 þ 2n12 n1 n2 ; a ¼ A11 n21 þ A22 n22 þ A33 n23 þ 2A12 n1 n2 ; c1 ¼ k4 n21 d1 þ k4 n22 d4 þ k4 n23 d6 þ 2n1 n2 k4 d2 ; 2
0 22 22 22 22 22 22 22 22 2 22 c2 ¼ ðq 12 d2 þ q22 d4 Þ; d2 ¼ q12 q33 ; d6 ¼ q11 q22 ðq12 Þ ; F 12 ¼ ðq12 n3 þ q33 n1 n2 Þk4 ; 0 0 0 22 22 22 22 2 22 22 2 22 F 13 ¼ ðq12 n2 n3 þ q22 n1 n3 Þk4 ; F 23 ¼ ðq11 n2 n3 q12 n1 n3 Þk4 ; F 33 ¼ ðq 22 n1 q11 n2 þ 2q12 n1 n2 Þk4 .
2 mm 222 2 mm, 222
e01 ¼ ðe15 þ e31 Þn1 n3 ; e02 ¼ ðe24 þ e32 Þn2 n3 ; e03 ¼ e15 n21 þ e24 n22 þ e33 n23 ; nk ¼ e3 n3 nk , e01 ¼ ðe25 þ e36 Þn2 n3 ; e02 ¼ ðe14 þ e36 Þn1 n3 ; e03 ¼ ðe25 þ e14 Þn1 n2 ; nk ¼ 0. P 11 ¼ c11 n21 þ c66 n22 þ c55 n23 ; P 12 ¼ ðc12 þ c66 Þn1 n2 ; P13 ¼ ðc13 þ c55 Þn1 n3 ; P 22 ¼ c66 n21 þ c22 n22 þ c44 n23 ; P 23 ¼ ðc23 þ c44 Þn2 n3 ; P33 ¼ c55 n21 þ c44 n22 þ c33 n23 ; n0 ¼ n11 n21 þ n22 n22 þ n33 n23 ; n0 ¼ n11 n21 þ n22 n22 þ n33 n23 ; a ¼ A11 n21 þ A22 n22 þ A33 n23 ; c1 ¼ k4 n21 d1 þ k4 n22 d4 þ k4 n23 d6 ; 22 22 22 0 22 c2 ¼ q 22 d4 ; d2 ¼ 0; d6 ¼ q11 q22 ; F 12 ¼ q33 n1 n2 k4 ;
2, 222, m, 2 mm 4
0 0 22 22 22 2 22 2 F 013 ¼ q 22 n1 n3 k4 ; F 23 ¼ q11 n2 n3 k4 ; F 33 ¼ ðq22 n1 q11 n2 Þk4 . 0 0 22 22 22 22 2 22 q 33 ; d4 ¼ q 11 q 33 ; d3 ¼ d5 ¼ 0; F 11 ¼ ðq 22 22 2 22 2 22 2 d1 ¼ q 33 n2 q22 n3 Þk4 ; F 22 ¼ ðq33 n1 q11 n3 Þk4 .
P 11 ¼ c11 n21 þ c66 n22 þ c44 n23 þ 2c16 n1 n2 ; P12 ¼ c16 n21 c16 n22 þ ðc12 þ c66 Þn1 n2 ; P 13 ¼ ðc13 þ c44 Þn1 n3 ; P22 ¼ c66 n21 þ c11 n22 þ c44 n23 2c16 n1 n2 ; P 23 ¼ ðc13 þ c44 Þn2 n3 ; P33 ¼ c44 n21 þ c44 n22 þ c33 n23 ; e01 ¼ ðe15 þ e31 Þn1 n3 e14 n2 n3 ; e02 ¼ ðe15 þ e31 Þn2 n3 þ e14 n1 n3 ; e03 ¼ e15 n21 þ e15 n22 þ e33 n23 ; nk ¼ e3 n3 nk .
32
P 11 ¼ c11 n21 þ c66 n22 þ c44 n23 þ 2c14 n2 n3 ; P12 ¼ ðc12 þ c66 Þn1 n2 þ 2c14 n1 n3 ; P 13 ¼ 2c14 n1 n2 þ ðc13 þ c44 Þn1 n3 ; P 22 ¼ c66 n21 þ c11 n22 þ c44 n23 2c14 n2 n3 ; P 23 ¼ c14 n21 c14 n22 þ ðc13 þ c44 Þn2 n3 ; P 33 ¼ c44 n21 þ c44 n22 þ c33 n23 ; e01 ¼ e11 n21 e11 n22 e14 n2 n3 ; e02 ¼ 2e11 n1 n2 þ e14 n1 n3 ; e03 ¼ 0; nk ¼ 0.
Note: In all the crystal classes, the expressions for ~fi ði ¼ 1; 2; 3Þ can be deduced from those of e0i by replacing eiJ ðJ ¼ 1; 2; . . . ; 6Þ with fiJ . Similarly n0k ðk ¼ 1; 2; 3Þ can be deduced from nk by replacing ek with ~fk . For 6 mm class the results are obtained from 2 mm class by equating quantities with subscripts 22, 24, 32, 55 by the quantities with subscripts 11, 15, 31, 44 respectively. For 43m class, the results are obtained from 222 class by equating quantities with subscripts 22, 23, 25, 33, 36, 55, 66 by the quantities with subscripts 11, 13, 14, 11, 14, 44, 44 respectively. The expressions of n0 ; n0 ; a; c1 ; c2 ; di ði ¼ 1; 2; . . . ; 6Þ; F ij ði; j ¼ 1; 2; 3Þ for the crystal classes 4 and 32 are same as of the crystal class 6 mm.
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