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Bleustein-Gulyaev waves in strained piezoelectric ceramics
Mechanics Research Communications.
Vol. 28, No. 6, pp. 679~83.2001
Copyright
0 2001 Elsevier Science Ltd
F’tinted in the USA.
All tights reserved
0093-6413/0l/$see
front matter
PII: SOO93-6413(02)00219-7
BLEUSTEIN-GULYAEV WAVES IN STRAINED PIEZOELECTRIC CERAMICS
J. S. Yang Department of Engineering Mechanics University of Nebraska, Lincoln, NE 68588
(Received 12 October 1999; acceptedfor print 16 November 2001)
Introduction
Bleustein-Gulyaev wave [l] is a piezoelectric surface wave that can propagate in polarized ceramics with only onedisplacement component and has no elastic counterpart. This type of waves has been studied extensively with wide applications in surface acoustic wave devices [2]. In device applications, mechanical, electrical, or thermal biasing fields may exit which can cause many undesirable effects. On the other hand, the effect of biasing fields on surface waves can be used to measure these biasing fields. This is the foundation of many sensors. The effect of biasing fields can be studied by the linear theory for small fields superposed on a finite bias, in which the biasing fields appear in the coefficients of the linear equations for the small incremental fields. For isotropic elastic bodies the effect of biasing mechanical fields on elastic surface waves was studied in [3] and many later papers (see [4]). For piezoelectric materials the theory for small fields superposed on a bias was developed in [5]. The theory has been used to analyze the effect of biasing mechanical and thermal fields on bulk and surface wave quartz resonators [6]. Since quartz is a materialwith weak piezoelectric coupling, in the analysis of the vibration characteristics of quartz resonators the electric field is usually neglected and only small biasing mechanical fields are considered so that a perturbation procedure can be used. Coupled electromechanical analysis for surface piezoelectric waves under finite biasing fields does not seem to have been reported. In this paper, the propagation of B-G waves in ceramics under finite biasing mechanical fields is studied. The theory of small electromechanical fields superposed on a finite bias is employed. It is shown that under finite extensional biasing fields a onedisplacement component surface wave can still propagate and is governed by two differential equations. The equations are solved analytically. The frequency equation that determines the wave speed is obtained. Some observations are made and a few special cases are discussed. The results are potentially useful in determining nonlinear material constants of ceramics by experiments.
679
680
J. S. YANG
Polarized Ceramics
Polarized ceramics possess a polarization vector but the electric field is already neutralized [5]. It is transversely isotropic (comm) and exhibits electromechanical coupling. We choose the reference state to be free from any mechanical fields. The constitutive relations of polarized ceramics are characterized by an energy density function C [5,6] of ten invariants [7] E= E(I,,I,,**.,I,,), I, =a-E.a,
I, =trE,
I, = trE2,
I, =W.W,
I, = trE3,
I, = We E . W,
I:, =8-W,
(1) I., =a-E*.a,
I, =a-E.W+W.E-a,
(2)
I,, =a.E2.W+W.E2-a,
where a is a vector along the poling direction, E is the Lagrangian finite strain tensor, and W the material electric field [5]. In the following, the above energy density will be understood to be the total energy density whose derivatives generate the total stress tensor (including Maxwell’s) and the electric displacement vector [8]
where the lowercase index i is summed form one to ten and Cartesian tensor notation has been used for capital indices. From TKL the first Piola-Kirchhoff stress tensor can be obtained which satisfies the equation of motion. The equation of electrostatics is satisfied by DK [5,6].
The Finite Biasing Field
Consider a ceramic half space occupying Xz < 0 in the reference configuration. The ceramic is poled in the Xs direction. The surface X2 = 0 is electroded and grounded with vanishing electric potential. Through a finite, static biasing stretching deformation 4, = AX,, 6, =&X2, 4; =&X, 2 (4) the material particle with referential coordinates XK is displaced to & which will be called the intermediate state for which Greek tensor indices are used. Eq. (4) produces a uniform biasing strain field E”. During the biasing deformation the electric potential is identical to zero which satisfies the electric boundary condition and implies a vanishing biasing electric field W” = 0. Then it can be verified from (3) and (2) that the only non-vanishing components of the biasing TA and 0: are T,:, Ti , TG and 0,” which are function of ill, &, ds and linear and nonlinear material parameters. This further implies that that the non-vanishing components of the first Piola-Kirchhoff stress tensor are (l+di) T,:, (l+&)Ti , and (I+A3) T,:. Since all these components are constants, the equilibrium equation and the equation of electrostatics (zero divergence of D, ) are satisfied. At X2 = 0, a normal traction of (l+&) TG is needed to satisfy the
681
WAVES IN STRAINED PIEZOELECTRIC CERAMICS
mechanical boundary conditions. The special case of a traction free surface can be obtained by setting T$ = 0 which represents a relation of hi, AZ,and 23. We denote Fun =
d2C
aE,aE,
& E.
w.
=-
)
a*c a5 aw,aE,, wo’ Efi=-aw,aw, EO
EO
wo.
(5)
These are also functions of 11, &, ;13and material parameters. Lengthy algebra shows that the tensors defined in (5) possess the symmetry of class 2mrn. This should not be surprising in view of the biasing fields applied. When the biasing fields vanish, these tensors have aomm symmetry. They reduce to the linear material constants of polarized ceramics when C is quadratic in E and W.
The Small Incremental Field
On top of the above biasing fields, small, dynamic mechanical displacement field un and electric field 4 (the electric potential) are superimposed. A material particle moves from & to the final state & + ua. Constitutive relations for the incremental stress and electric displacement are [5,6] T,: =&E:,
-Zm,W;,
0;
=&,E;,
+&W;,
(6)
where the incremental strain and electric field are defined as E:, = (5&K%, + &?,‘$,K ) 123 w; = -#,, * The incremental Piola-Kirchhoff stress tensor is given by
(7)
K:, = %& +5&L * The incremental equations of motion and electrostatics take the form
(8)
&.K = P&, , D:., = 0, (9) where pi is the reference mass density and all equations are written with respect to the referential coordinates X. We want to study the possibility of B-G type incremental waves with one displacement component u&xi,xz,t) and &i,xz,r). Then, for ceramics under the biasing fields of (4), from (6) we obtain the following nontrivial components Ki3 = ck,,
+ e$.,,
0,’ = ek,., -Q,,,
K:, = c;.+%,~+ e;.,k2, G, = 4 &%,, G, =
22 K4%.*
0: = e~,~,,, -%24.2 y
(10)
+ ~5~,, 1, + ~244,2),
where cl, = T,: + &*&, I$, = TA + @.,, , e; = A,Z,5, e;., = A.,Z2;,, (11) and the compact matrix notation [9] for material tensors has been used. Substitution of (10) into (9) yields . css~3,11 + w43,22 + &!,, + h?22 = PO%9 . . w3,11 + ez4u3,,,- w4,,, - E22Q1,22 = 0. l
(12)
J. S. YANG
682
We note that the difference between (12) and the equations that govern B-G waves in a medium without biasing fields [l] is due to the change of symmetry from oomm to the effective symmetry of 2mm under the bias given by (4). The boundary conditions at X2 = 0 lefl to be satisfied are Ki, = c;u,,, + e;@,z = 0, 4 = 0.
(13)
Surface Wave Solution and Discussion
We look for surface waves propagating in the Xt direction in the form , / = Be7hei(&-duc) Uj = Aeflz ,iC5%-M ,
(14) where A, B, 4, 77and o are constants. Substituting (14) into (12), requiring nontrivial solutions for A and B, one obtains
- c;,r2 + c;7J2 -eX’ +eh2 = POW2 -e;{'
+
e;,v2
4,r2
-E22712
o
(15) .
In the special case when there are no biasing fields, (15) has two positive real roots for 77which represent exponential decay from the surface. When biasing fields are present, (15) may have conjugate pairs of complex roots for 7. In that case the two roots with positive real parts will be chosen. Denoting those two roots by 77(‘)and v(2)which are functions of 4 and w, the desired solution to (12) can be written as Us = c(r)[E,,52 _ E22(@ )2 ]e#‘x~ei(FY1-~) + ~(2)[~,,~2 _ E22($))2 ]edt)%ei(fll-~), 4 = cW[e;542 _ e;,(t1(l))2]e”(‘)x~ei(ZXI-ow) + ~(2)[~;,32 _ e;,(TI(2))2 ]es”‘X2ei(Wk-W,
(16)
where c”’ and ti2’ are undetermined constants. Substituting (16) into (13), for nontrivial solutions of c”’ and d”, we obtain the following equation {c;[E,,c2 -~22(~c))2]r+‘) +e;[etr’ -eI,(f1”‘)2]Tl”‘}[e;,r2 -e;,(r+“)‘] (17) - {c;[E,,c2 -&22(+2))2]r+2) +e&[e;,e2 -e;(r1’2’)2]77(2))[e~~52-e;(#“)2] = 0. Equation (17) determines relations of w versus 5 (dispersion relations). It can also be written as an equation for the wave speed v = ol{. Examination of the structures of (17) and (15) shows that waves determined by (17) are not dispersive (o is linear in 5 or v is a constant, not depending on 5). In the special case when biasing fields are not present, there exists a real positive speed for B-G waves. In the case when the biasing fields are isotropic in the X,-X2 plane (21 = 22, which also implies T,; = T$ = To), the strained body is effectively transversely isotropic. Then the mathematical structure of (12) and (13) becomes the same as those in [ 11, and a wave speed solution in the following form can be obtained [l] v2 = e&l PO
I (eh2 E,,CL
W Ill-
$&~,,j;l~;‘“!‘12~~ &llC44
Eq. (18) is in fact a very complicated function of the biasing fields and material parameters. When the material is linear, the dependence on the biasing fields becomes simple and explicit
WAVES IN STRAINED PIEZOELECTRIC CERAMICS
g =
U-O +4c44) [I
PO
+
683
1) (19) .,,(:&c‘y- )2;;n;e;, +A&,J $
(TO
II
2
+A2c
344
E,,(TO
where ~44, eis and ~11 are the usual linear elastic, piezoelectric, and dielectric constants of ceramics [9]. Eqs. (17) and (18) can not be further explored at this stage due to lack of nonlinear material constants of ceramics. In fact, the main purpose of this work is to provide results that can be used to measure nonlinear material constants experimentally. Since (17) or (18) depends on nonlinear material constants, combinations of these constants can be determined by measuring wave speeds under various biasing fields [lo]. For polarized ceramics, there are twenty five third-order nonlinear material constants [ 111.To determine all of them, in addition to (17) or (1 S), it will be more convenient to have solutions of other static or dynamic responses for ceramic bodies under various biasing fields. In general, solutions of this nature will be complicated. Another possible application of (17) or (18) is in the measurement of initial stresses through their effects on wave speeds [ 121.
Acknowledgement
This work is supported by a Faculty Summer Research Fellowship for 1999 from the Research Council of University of Nebraska-Lincoln with grant number LWT/l l-l 54-91701.
References
1. J. L. Bleustein, Applied Physics Letters 13,412 (1968). 2. V. Z. Patton and B. A. Kudryavtsev, Electromagnetoelasticity. Gordon and Breach, New York (1988). 3. H. Hayes and R. S. Rivlin, Arch. Ration. Mech. Analysis 8,358 (1961). 4. D. Iesan, Prestressed Bodies. Longman, Harlow, Essex, UK (1989). 5. J. C. Baumhauer and H. F. Tiersten, J. Acoust. Sot. Am. 54, 1017 (1973). 6. H. F. Tiersten, Int. J. Engng Sci. 33, 2239 (1995). 7. Q. S. Zheng, Int. J. Engng Sci. 31, 1399 (1993). 8. J. S. Yang and R. C. Batra, International Journal of Nonlinear Mechanics 30,719 (1995). 9. B. A. Auld, Acoustic Fields and Waves in Solids. Wiley, New York (1973). 10. Y. Cho and K. Yamanouchi, J. Appl. Phys. 61,875 (1987). 11. J. S. Yang and R. C. Batra, J. Acoust. Sot. Am. 97,280 (1995). 12. Y.-H. Pao, W. Sachse and H. Fukuoda, in Physical Acoustics Vol. XVII. W. P. Mason and R. N. Thurston ed., Academic Press, Orlando, Florida, 62 (1984).
Vol. 28, No. 6, pp. 679~83.2001
Copyright
0 2001 Elsevier Science Ltd
F’tinted in the USA.
All tights reserved
0093-6413/0l/$see
front matter
PII: SOO93-6413(02)00219-7
BLEUSTEIN-GULYAEV WAVES IN STRAINED PIEZOELECTRIC CERAMICS
J. S. Yang Department of Engineering Mechanics University of Nebraska, Lincoln, NE 68588
(Received 12 October 1999; acceptedfor print 16 November 2001)
Introduction
Bleustein-Gulyaev wave [l] is a piezoelectric surface wave that can propagate in polarized ceramics with only onedisplacement component and has no elastic counterpart. This type of waves has been studied extensively with wide applications in surface acoustic wave devices [2]. In device applications, mechanical, electrical, or thermal biasing fields may exit which can cause many undesirable effects. On the other hand, the effect of biasing fields on surface waves can be used to measure these biasing fields. This is the foundation of many sensors. The effect of biasing fields can be studied by the linear theory for small fields superposed on a finite bias, in which the biasing fields appear in the coefficients of the linear equations for the small incremental fields. For isotropic elastic bodies the effect of biasing mechanical fields on elastic surface waves was studied in [3] and many later papers (see [4]). For piezoelectric materials the theory for small fields superposed on a bias was developed in [5]. The theory has been used to analyze the effect of biasing mechanical and thermal fields on bulk and surface wave quartz resonators [6]. Since quartz is a materialwith weak piezoelectric coupling, in the analysis of the vibration characteristics of quartz resonators the electric field is usually neglected and only small biasing mechanical fields are considered so that a perturbation procedure can be used. Coupled electromechanical analysis for surface piezoelectric waves under finite biasing fields does not seem to have been reported. In this paper, the propagation of B-G waves in ceramics under finite biasing mechanical fields is studied. The theory of small electromechanical fields superposed on a finite bias is employed. It is shown that under finite extensional biasing fields a onedisplacement component surface wave can still propagate and is governed by two differential equations. The equations are solved analytically. The frequency equation that determines the wave speed is obtained. Some observations are made and a few special cases are discussed. The results are potentially useful in determining nonlinear material constants of ceramics by experiments.
679
680
J. S. YANG
Polarized Ceramics
Polarized ceramics possess a polarization vector but the electric field is already neutralized [5]. It is transversely isotropic (comm) and exhibits electromechanical coupling. We choose the reference state to be free from any mechanical fields. The constitutive relations of polarized ceramics are characterized by an energy density function C [5,6] of ten invariants [7] E= E(I,,I,,**.,I,,), I, =a-E.a,
I, =trE,
I, = trE2,
I, =W.W,
I, = trE3,
I, = We E . W,
I:, =8-W,
(1) I., =a-E*.a,
I, =a-E.W+W.E-a,
(2)
I,, =a.E2.W+W.E2-a,
where a is a vector along the poling direction, E is the Lagrangian finite strain tensor, and W the material electric field [5]. In the following, the above energy density will be understood to be the total energy density whose derivatives generate the total stress tensor (including Maxwell’s) and the electric displacement vector [8]
where the lowercase index i is summed form one to ten and Cartesian tensor notation has been used for capital indices. From TKL the first Piola-Kirchhoff stress tensor can be obtained which satisfies the equation of motion. The equation of electrostatics is satisfied by DK [5,6].
The Finite Biasing Field
Consider a ceramic half space occupying Xz < 0 in the reference configuration. The ceramic is poled in the Xs direction. The surface X2 = 0 is electroded and grounded with vanishing electric potential. Through a finite, static biasing stretching deformation 4, = AX,, 6, =&X2, 4; =&X, 2 (4) the material particle with referential coordinates XK is displaced to & which will be called the intermediate state for which Greek tensor indices are used. Eq. (4) produces a uniform biasing strain field E”. During the biasing deformation the electric potential is identical to zero which satisfies the electric boundary condition and implies a vanishing biasing electric field W” = 0. Then it can be verified from (3) and (2) that the only non-vanishing components of the biasing TA and 0: are T,:, Ti , TG and 0,” which are function of ill, &, ds and linear and nonlinear material parameters. This further implies that that the non-vanishing components of the first Piola-Kirchhoff stress tensor are (l+di) T,:, (l+&)Ti , and (I+A3) T,:. Since all these components are constants, the equilibrium equation and the equation of electrostatics (zero divergence of D, ) are satisfied. At X2 = 0, a normal traction of (l+&) TG is needed to satisfy the
681
WAVES IN STRAINED PIEZOELECTRIC CERAMICS
mechanical boundary conditions. The special case of a traction free surface can be obtained by setting T$ = 0 which represents a relation of hi, AZ,and 23. We denote Fun =
d2C
aE,aE,
& E.
w.
=-
)
a*c a5 aw,aE,, wo’ Efi=-aw,aw, EO
EO
wo.
(5)
These are also functions of 11, &, ;13and material parameters. Lengthy algebra shows that the tensors defined in (5) possess the symmetry of class 2mrn. This should not be surprising in view of the biasing fields applied. When the biasing fields vanish, these tensors have aomm symmetry. They reduce to the linear material constants of polarized ceramics when C is quadratic in E and W.
The Small Incremental Field
On top of the above biasing fields, small, dynamic mechanical displacement field un and electric field 4 (the electric potential) are superimposed. A material particle moves from & to the final state & + ua. Constitutive relations for the incremental stress and electric displacement are [5,6] T,: =&E:,
-Zm,W;,
0;
=&,E;,
+&W;,
(6)
where the incremental strain and electric field are defined as E:, = (5&K%, + &?,‘$,K ) 123 w; = -#,, * The incremental Piola-Kirchhoff stress tensor is given by
(7)
K:, = %& +5&L * The incremental equations of motion and electrostatics take the form
(8)
&.K = P&, , D:., = 0, (9) where pi is the reference mass density and all equations are written with respect to the referential coordinates X. We want to study the possibility of B-G type incremental waves with one displacement component u&xi,xz,t) and &i,xz,r). Then, for ceramics under the biasing fields of (4), from (6) we obtain the following nontrivial components Ki3 = ck,,
+ e$.,,
0,’ = ek,., -Q,,,
K:, = c;.+%,~+ e;.,k2, G, = 4 &%,, G, =
22 K4%.*
0: = e~,~,,, -%24.2 y
(10)
+ ~5~,, 1, + ~244,2),
where cl, = T,: + &*&, I$, = TA + @.,, , e; = A,Z,5, e;., = A.,Z2;,, (11) and the compact matrix notation [9] for material tensors has been used. Substitution of (10) into (9) yields . css~3,11 + w43,22 + &!,, + h?22 = PO%9 . . w3,11 + ez4u3,,,- w4,,, - E22Q1,22 = 0. l
(12)
J. S. YANG
682
We note that the difference between (12) and the equations that govern B-G waves in a medium without biasing fields [l] is due to the change of symmetry from oomm to the effective symmetry of 2mm under the bias given by (4). The boundary conditions at X2 = 0 lefl to be satisfied are Ki, = c;u,,, + e;@,z = 0, 4 = 0.
(13)
Surface Wave Solution and Discussion
We look for surface waves propagating in the Xt direction in the form , / = Be7hei(&-duc) Uj = Aeflz ,iC5%-M ,
(14) where A, B, 4, 77and o are constants. Substituting (14) into (12), requiring nontrivial solutions for A and B, one obtains
- c;,r2 + c;7J2 -eX’ +eh2 = POW2 -e;{'
+
e;,v2
4,r2
-E22712
o
(15) .
In the special case when there are no biasing fields, (15) has two positive real roots for 77which represent exponential decay from the surface. When biasing fields are present, (15) may have conjugate pairs of complex roots for 7. In that case the two roots with positive real parts will be chosen. Denoting those two roots by 77(‘)and v(2)which are functions of 4 and w, the desired solution to (12) can be written as Us = c(r)[E,,52 _ E22(@ )2 ]e#‘x~ei(FY1-~) + ~(2)[~,,~2 _ E22($))2 ]edt)%ei(fll-~), 4 = cW[e;542 _ e;,(t1(l))2]e”(‘)x~ei(ZXI-ow) + ~(2)[~;,32 _ e;,(TI(2))2 ]es”‘X2ei(Wk-W,
(16)
where c”’ and ti2’ are undetermined constants. Substituting (16) into (13), for nontrivial solutions of c”’ and d”, we obtain the following equation {c;[E,,c2 -~22(~c))2]r+‘) +e;[etr’ -eI,(f1”‘)2]Tl”‘}[e;,r2 -e;,(r+“)‘] (17) - {c;[E,,c2 -&22(+2))2]r+2) +e&[e;,e2 -e;(r1’2’)2]77(2))[e~~52-e;(#“)2] = 0. Equation (17) determines relations of w versus 5 (dispersion relations). It can also be written as an equation for the wave speed v = ol{. Examination of the structures of (17) and (15) shows that waves determined by (17) are not dispersive (o is linear in 5 or v is a constant, not depending on 5). In the special case when biasing fields are not present, there exists a real positive speed for B-G waves. In the case when the biasing fields are isotropic in the X,-X2 plane (21 = 22, which also implies T,; = T$ = To), the strained body is effectively transversely isotropic. Then the mathematical structure of (12) and (13) becomes the same as those in [ 11, and a wave speed solution in the following form can be obtained [l] v2 = e&l PO
I (eh2 E,,CL
W Ill-
$&~,,j;l~;‘“!‘12~~ &llC44
Eq. (18) is in fact a very complicated function of the biasing fields and material parameters. When the material is linear, the dependence on the biasing fields becomes simple and explicit
WAVES IN STRAINED PIEZOELECTRIC CERAMICS
g =
U-O +4c44) [I
PO
+
683
1) (19) .,,(:&c‘y- )2;;n;e;, +A&,J $
(TO
II
2
+A2c
344
E,,(TO
where ~44, eis and ~11 are the usual linear elastic, piezoelectric, and dielectric constants of ceramics [9]. Eqs. (17) and (18) can not be further explored at this stage due to lack of nonlinear material constants of ceramics. In fact, the main purpose of this work is to provide results that can be used to measure nonlinear material constants experimentally. Since (17) or (18) depends on nonlinear material constants, combinations of these constants can be determined by measuring wave speeds under various biasing fields [lo]. For polarized ceramics, there are twenty five third-order nonlinear material constants [ 111.To determine all of them, in addition to (17) or (1 S), it will be more convenient to have solutions of other static or dynamic responses for ceramic bodies under various biasing fields. In general, solutions of this nature will be complicated. Another possible application of (17) or (18) is in the measurement of initial stresses through their effects on wave speeds [ 121.
Acknowledgement
This work is supported by a Faculty Summer Research Fellowship for 1999 from the Research Council of University of Nebraska-Lincoln with grant number LWT/l l-l 54-91701.
References
1. J. L. Bleustein, Applied Physics Letters 13,412 (1968). 2. V. Z. Patton and B. A. Kudryavtsev, Electromagnetoelasticity. Gordon and Breach, New York (1988). 3. H. Hayes and R. S. Rivlin, Arch. Ration. Mech. Analysis 8,358 (1961). 4. D. Iesan, Prestressed Bodies. Longman, Harlow, Essex, UK (1989). 5. J. C. Baumhauer and H. F. Tiersten, J. Acoust. Sot. Am. 54, 1017 (1973). 6. H. F. Tiersten, Int. J. Engng Sci. 33, 2239 (1995). 7. Q. S. Zheng, Int. J. Engng Sci. 31, 1399 (1993). 8. J. S. Yang and R. C. Batra, International Journal of Nonlinear Mechanics 30,719 (1995). 9. B. A. Auld, Acoustic Fields and Waves in Solids. Wiley, New York (1973). 10. Y. Cho and K. Yamanouchi, J. Appl. Phys. 61,875 (1987). 11. J. S. Yang and R. C. Batra, J. Acoust. Sot. Am. 97,280 (1995). 12. Y.-H. Pao, W. Sachse and H. Fukuoda, in Physical Acoustics Vol. XVII. W. P. Mason and R. N. Thurston ed., Academic Press, Orlando, Florida, 62 (1984).