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Acta Mechanica 161, 1–16 (2003) DOI 10.1007/s00707-002-0986-x
Acta Mechanica Printed in Austria
Thermally-induced wave propagation in a piezoelectric plate F. Ashida, Shimane, Japan, and T. R. Tauchert, Lexington, Kentucky Received October 3, 2002 Published online: February 10, 2003 Ó Springer-Verlag 2003
Summary. The Laplace transform technique is utilized to obtain the exact solution for thermally-induced wave propagation in a circular piezoelectric plate. The temperature is assumed to satisfy the one-dimensional heat conduction equation that includes a relaxation time. Displacement and electric potential functions are introduced in order to solve the equations of motion and electrodynamics. Numerical results obtained for both a PZT-5A plate and a non-piezoelectric plate illustrate the effects of relaxation time and piezoelasticity on displacement and stress histories.
1 Introduction Numerous papers have dealt with dynamic problems involving electromechanical structures. Most of the investigations have treated vibration problems, such as detection and/or control of harmonic vibrations in various host structures by means of attached piezoelectric sensors and/ or actuators. References [1]–[6], for example, consider piezo control of structural vibration under isothermal conditions, while [7]–[11] consider vibration suppression of structures exposed to thermal environments. Although the literature on harmonic vibrations of piezoelectric structures is plentiful, little attention has been given to wave propagation in piezoelectric materials. One study in this area is that of Wang and Huang [12], who analyzed an isothermal problem of wave propagation in an electro-elastic structure when the host structure is excited by means of piezoelectric actuators. It is expected that piezoceramic thin single crystal films of nano-order thickness will be put to practical use in the near future. When thermal loads act on such thin films, effects of both a relaxation time of heat flux and inertia effects in the thermoelastic field may be significant and should not be neglected. Inclusion of a relaxation time is necessary in order to eliminate the paradox of infinite velocity of a thermal wave, as is predicted by classical heat conduction theory. The importance of considering a relaxation time in the analysis of wave propagation in thermoelastic bodies is well documented (see Chandrasekharaiah [13] and Ignaczak [14], [15]). The present paper deals with the dynamic thermoelastic problem of a thin piezoelectric plate of crystal class 6mm. A relaxation time of heat flux, namely that attributed to Lord and Shulman [16], is taken into account. One boundary surface of the plate is exposed to a sudden uniform ambient temperature, whereas the other boundary surface is kept at zero temperature. An exact solution to this problem is obtained by employing the Laplace transform technique.
2
F. Ashida and T. R. Tauchert
Numerical calculations have been carried out for a thin PZT-5A plate, and the results are illustrated graphically.
2 Problem statement Consider a thin circular piezoelectric plate having material symmetry properties of crystal class 6 mm. The radius and thickness of the plate are a and b, respectively. It is assumed that initially the temperature throughout the plate is zero, and that the plate is free of deformation, stress, electric displacement and electric potential.
2.1 Temperature field Suppose that at time t ¼ 0 the top surface of the plate (z ¼ b) is exposed to a uniform ambient temperature Tc . The lower surface (z ¼ 0) is maintained at zero temperature, and the cylindrical edge (r ¼ a) is thermally insulated. In this case, the thermal initial and boundary conditions are expressed as T ¼ 0;
T;t ¼ 0 at t ¼ 0;
ð1Þ
T;z þ hT ¼ hTc
on
z ¼ b;
ð2Þ
T¼0
on
z ¼ 0;
ð3Þ
T;r ¼ 0
on
r ¼ a;
ð4Þ
where T denotes temperature, and h is the relative surface heat transfer coefficient. The temperature is assumed to satisfy the one-dimensional heat conduction equation which includes a relaxation time t0 , namely 1 k2 T;zz ¼ ðT;t þ t0 T;tt Þ: j
ð5Þ
Here k2 ¼ kz =kr in which kz and kr are coefficients of thermal conductivity, and j represents thermal diffusivity.
2.2 Elastic and electric field In determining the elastic and electric fields induced by the suddenly applied ambient temperature Tc , it is assumed that the cylindrical edge of the plate is smoothly constrained against radial deformation and free of electric charge. Then ur ¼ 0;
rrz ¼ 0;
ð6Þ
Er ¼ 0;
Dr ¼ 0;
ð7Þ
while all other response quantities are functions only of the spatial coordinate z and time t; in particular rrr ¼ rhh ¼ c13 uz;z e1 Ez b1 T;
ð8Þ
rzz ¼ c33 uz;z e3 Ez b3 T;
ð9Þ
Dz ¼ e3 uz;z þ g3 Ez þ p3 T;
ð10Þ
3
Thermally-induced wave propagation
where ui are elastic displacements, rik are stresses, Ei are electric field intensities, Di are electric displacements, cik are elastic moduli, ei are piezoelectric coefficients, bi are stress-temperature coefficients, g3 is a dielectric permittivity, and p3 is a pyroelectric constant. In this case, the elastic field is governed by the equation of motion rzz;z ¼ quz;tt ;
ð11Þ
where q denotes mass density, and the electric field satisfies the equation of electrodynamics Dz;zt ¼ 0:
ð12Þ
The upper and lower faces of the plate are considered to be free of both traction and electric charge. In this case the initial and boundary conditions are expressed as uz ¼ uz ;t ¼ 0 at t ¼ 0;
ð13Þ
Ez ¼ 0
at t ¼ 0;
ð14Þ
rzz ¼ 0
on z ¼ 0; b;
ð15Þ
Dz ¼ 0
on z ¼ 0; b:
ð16Þ
3 Analysis 3.1 Temperature field To determine the temperature field that satisfies the heat conduction Eq. (5) and initial conditions (1), we apply the Laplace transform Z 1 T ¼ Tept dt: ð17Þ 0
Equations (5), (2) and (3) become p k2 T;zz ¼ ð1 þ t0 pÞT ; j hTc T;z þ hT ¼ on z ¼ b; p T ¼ 0
on z ¼ 0:
ð18Þ ð19Þ ð20Þ
The solution to (18) that satisfies boundary condition (20) is taken to be rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pð1 þ t0 pÞ z : T ¼ A sinh j k
ð21Þ
Substitution of (21) into (19) gives k hT qffiffiffiffiffiffiffiffiffiffiffiffiffiffi c qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; A ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1þt0 pÞ pð1þt0 pÞ b pð1þt0 pÞ b p cosh j j j k þ k h sinh k
ð22Þ
in which case (21) becomes qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pð1þt0 pÞ z k hTc sinh j k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : T ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1þt0 pÞ pð1þt0 pÞ b pð1þt0 pÞ b p cosh j j j k þ k h sinh k
ð23Þ
4
F. Ashida and T. R. Tauchert
Inversion of T is performed using the method of residues, with the result that
1 X z am t 1 am t t þ Am H a2m cos sin T ¼ A0 þ e 2t0 b 2t0 am 2t0 m¼1 em t 1 em t c z þ sin m ; þ Hðe2m Þ sinh 2t0 em 2t0 k
ð24Þ
where HðxÞ is the Heaviside unit step function and A0 ¼
am ¼
hbTc ; 1 þ hb
Am ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4jt0 c2m 1;
2 k hTc n o ; cm ð1 þ hbÞ cos cmk b cmk b sin cmk b em ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4jt0 c2m ;
ð25Þ
ð26Þ
and cm are the roots of the equation cm b cm b þ k h sin ¼ 0: cm cos k k
ð27Þ
3.2 Elastic and electric fields In order to solve the equations of motion (11) and electrodynamics (12), we introduce a displacement potential X and an electric potential U defined by the relations uz ¼ X;z ;
ð28Þ
Ez ¼ U;z :
ð29Þ
Substitution of (28) and (29) into (8)–(10) gives the following expressions for the stresses and electric displacement: rrr ¼ rhh ¼ c13 X;zz þ e1 U;z b1 T;
ð30Þ
rzz ¼ c33 X;zz þ e3 U;z b3 T;
ð31Þ
Dz ¼ e3 X;zz g3 U;z þ p3 T:
ð32Þ
By introducing (30)–(32) into (11) and (12), we obtain the equations governing the displacement and electric potentials, namely X;zz
1 X;tt ¼ n1 T; m2e
ð33Þ
U;z ¼ dX;zz þ n2 T; in which 1 e2 c33 þ 3 ; m2e ¼ q g3
ð34Þ
n1 ¼
g 3 b 3 e 3 p3 ; g3 c33 þ e23
n2 ¼
p3 ; g3
d¼
e3 : g3
ð35Þ
In this case the electric displacement Dz is identically zero and thus the boundary condition (16) has been satisfied. The aforementioned initial conditions, namely (13) and (14), imply that X ¼ X;t ¼ U ¼ 0
at t ¼ 0:
ð36Þ
Next applying the Laplace transform to (33) and (34) under the initial conditions (36) gives
Thermally-induced wave propagation
X;zz
p2 X ¼ n1 T ; m2e
5
ð37Þ
U;z ¼ dX;zz þ n2 T :
ð38Þ
Substituting (23) for T into (37) and (38) and then integrating gives rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pð1 þ t0 pÞ z pz þ E sinh X ¼ C sinh ; j k me rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pð1 þ t0 pÞ z pz U ¼ D cosh þ F cosh ; j k me
ð39Þ
ð40Þ
where C ¼ n1 khTc
p2
1 t0 1 þ jk2 jk2 m2e
(rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !)# pð1 þ t0 pÞ pð1 þ t0 pÞ b pð1 þ t0 pÞ b cosh þ k h sinh j j k j k " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 dð1 þ t0 pÞ 1 t0 1 pð1 þ t0 pÞ 1 t0 1 2 þ n2 k p hTc p p þ þ D¼ j j jk2 m2e jk2 m2e jk2 jk2 (rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !)# pð1 þ t0 pÞ pð1 þ t0 pÞ b pð1 þ t0 pÞ b cosh þ k h sinh ; j j k j k F¼
ð41Þ
d p E: me
By combining (11) and (28), the axial stress is expressed as rzz ¼ qX;tt :
ð42Þ
Taking the Laplace transform of this equation gives rzz ¼ qp2 X :
ð43Þ
Substituting (39) into (43) then yields ( rzz
2
¼ qp
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ) pð1 þ t0 pÞ z pz þ E sinh : C sinh j k me
ð44Þ
Applying boundary condition (15), written in the transform domain as rzz ¼ 0
on z ¼ 0; b;
ð45Þ
leads to the result sinh
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1þt0 pÞ b j k
E¼ sinh
pb me
C:
ð46Þ
6
F. Ashida and T. R. Tauchert
Substitution of (41) and (46) for the coefficients C; D; E and F into (39) and (40) gives ( rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ) pb pð1 þ t0 pÞ z pð1 þ t0 pÞ b pz sinh sinh X ¼ n1 khTc sinh sinh me j k j k me
" 1 t0 1 pb þ p sinh p2 2 2 2 me me jk jk (rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !)# pð1 þ t0 pÞ pð1 þ t0 pÞ b pð1 þ t0 pÞ b cosh þ kh sinh ; j j k j k
ð47Þ
and "
(rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pð1 þ t0 pÞ pb pð1 þ t0 pÞ z sinh U ¼ hTc n1 d cosh j me j k
kp sinh me
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! )
pð1 þ t0 pÞ b pz 1 t0 1 pb cosh p sinh p þ j k me me jk2 jk2 m2e
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !,rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# pð1 þ t0 pÞ z pð1 þ t0 pÞ þ n2 k cosh j k j 2
," (rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !)# pð1 þ t0 pÞ pð1 þ t0 pÞ b pð1 þ t0 pÞ b cosh þ kh sinh : p j j k j k
ð48Þ
Inversion of X and U by the method of residues leads to the results X¼
n 1 A0 3 ðz b2 zÞ 6b þe
2tt
0
1 X
Hða2m ÞF12m
m¼1
c z am t am t F1m sin sin m F2m cos 2t0 2t0 k
am t am t z am z þ F11m sin sinh cos þ F10m cos 2t0 2t0 2t0 me 2t0 me
am t am t z am z F10m sin cosh sin F11m cos 2t0 2t0 2t0 me 2t0 me þ
Hðe2m ÞK12m
c z em t em t K1m sinh sin m K2m cosh 2t0 2t0 k
em t em t z em z þ K11m sinh sinh cosh þ K10m cosh 2t0 2t0 2t0 me 2t0 me
em t em t z em z K11m cosh þ K10m sinh cosh sinh 2t0 2t0 2t0 me 2t0 me
1 X j¼1
G12j G13j cos ðme xj tÞ þ G14j sin ðme xj tÞ sin ðxj zÞ;
ð49Þ
Thermally-induced wave propagation
7
and U¼
A0 n 1 d n2 b2 t t ð3z2 b2 Þ þ z2 b2 þ hb z2 þ n2 jk2 t0 1 þ e t0 t0 b 6 2ð1 þ hbÞ 3
1 c z X dc F12m am t am t t e 2t0 F1m sin F2m cos cos m Hða2m Þ m 2t0 2t0 k k m¼1
dF12m am t am t þ ðam F11m F10m Þ cos ðF11m þ am F10m Þ sin 2t0 2t0 2t0 me z am z am t cos þ ðF11m þ am F10m Þ cos cosh 2t0 me 2t0 me 2t0 ) am t z am z sinh sin þ ðam F11m F10m Þ sin 2t0 2t0 me 2t0 me
c z n kAm am t 1 am t þ cos m þ 2 cos sin 2t0 am 2t0 k cm
c z dc K12m em t em t þ Hðe2m Þ m K1m sinh K2m cosh cos m 2t0 2t0 k k
dK12m em t em t ðK11m em K10m Þ sinh þ ðem K11m K10m Þ cosh 2t0 2t0 2t0 me z em z em t cosh þ ðK11m em K10m Þ cosh cosh 2t0 me 2t0 me 2t0
em t z em z sinh sinh ðem K11m K10m Þ sinh 2t0 2t0 me 2t0 me
n2 kAm em t 1 em t cm z þ cos þ cosh sinh 2t0 em 2t0 k cm d
1 X
xj G12j G13j cos ðme xj tÞ þ G14j sin ðme xj tÞ cos ðxj zÞ;
ð50Þ
j¼1
where xj ¼ jp=b, and the coefficients Fim ; Kim ði ¼ 1; 2; 10; 11; 12Þ and Gij ði ¼ 12; 13; 14Þ are defined in the Appendix. The transverse elastic displacement, found by substituting (49) into (28), becomes uz ¼
n 1 A0 ð3z2 b2 Þ 6b þe
2tt
0
1 X
Hða2m ÞF12m
m¼1
c z cm am t am t F2m cos F1m sin cos m 2t0 2t0 k k
1 am t am t þ ðF11m þ am F10m Þ sin ðF10m am F11m Þ cos þ 2t0 me 2t0 2t0 cosh
z 2t0 me
cos
am z 2t0 me
8
F. Ashida and T. R. Tauchert
1 am t am t ðF10m am F11m Þ sin ðF11m þ am F10m Þ cos 2t0 me 2t0 2t0
z 2t0 me
sinh
cos
sin
am z 2t0 me
þ Hðe2m ÞK12m
cm em t em t K2m cosh K1m sinh 2t0 2t0 k
c z 1 em t em t m þ ðem K10m K11m Þ sinh ðem K11m K10m Þ cosh k 2t0 me 2t0 2t0
z em z 1 em t cosh ðem K10m K11m Þ cosh cosh þ 2t0 me 2t0 me 2t0 me 2t0
em t z em z þ ðem K11m K10m Þ sinh sinh sinh 2t0 2t0 me 2t0 me
1 X
xj G12j G13j cos ðme xj tÞ þ G14j sin ðme xj tÞ cos ðxj zÞ:
ð51Þ
j¼1
Substitution of (24), (49) and (50) into (30) and (42) yields the following expressions for the elastic stresses:
ðc13 þ e1 dÞn1 rrr ¼ rhh ¼ b1 þ e1 n2 A0 z b e
2tt
0
1 X m¼1
ðc13 þ e1 dÞc2m F12m am t am t F F Hða2m Þ cos sin 2m 1m 2 2t0 2t0 k
c z am t 1 am t þ sin m cos sin 2t0 am 2t0 k
þðb1 e1 n2 ÞAm
ðc13 þ e1 dÞF12m ð2t0 me Þ2
F18m
am t am t z am z þ F17m sin sinh cos cos 2t0 2t0 2t0 me 2t0 me
am t am t z am z F18m sin cosh sin F17m cos 2t0 2t0 2t0 me 2t0 me
Hðe2m Þ
ðc13 þ e1 dÞc2m K12m em t em t K1m sinh K2m cosh 2t0 2t0 k2
þðb1 e1 n2 ÞAm
c z em t 1 em t þ sin m cosh sinh 2t0 em 2t0 k
ðc13 þ e1 dÞK12m ð2t0 me Þ2
em t em t z em z þ K17m sinh sinh cosh K18m cosh 2t0 2t0 2t0 me 2t0 me
em t em t z em z K17m cosh þ K18m sinh cosh sinh 2t0 2t0 2t0 me 2t0 me þ ðc13 þ e1 dÞ
1 X j¼1
x2j G12j G13j cos ðme xj tÞ þ G14j sin ðme xj tÞ sin ðxj zÞ;
ð52Þ
Thermally-induced wave propagation
9
1 q 2tt X am t am t cm z 2 0 F sin e Hða ÞF F cos sin 12m 19m 20m m 2t0 2t0 c 4t20 m¼1
am t am t z am z þ F17m sin sinh cos þ F18m cos 2t0 2t0 2t0 me 2t0 me
am t am t z am z F18m sin cosh sin F17m cos 2t0 2t0 2t0 me 2t0 me
em t em t c z þ Hðe2m ÞK12m K19m cosh K20m sinh sin m 2t0 2t0 k
em t em t z em z þ K18m cosh þ k17m sinh sinh cosh 2t0 2t0 2t0 me 2t0 me
em t em t z em z þ K18m sinh cosh sinh K17m cosh 2t0 2t0 2t0 me 2t0 me
rzz ¼
þ qm2e
1 X
x2j G12j G13j cos ðme xj tÞ þ G14j sin ðme xj tÞ sin ðxj zÞ;
ð53Þ
j¼1
where the coefficients Fim ; Kim ði ¼ 17; 18; 19; 20Þ are defined in the Appendix.
4 Numerical results Numerical calculations have been carried out for a PZT-5A plate. The material constants are taken to be: q ¼ 7750 kg m3 ; kr ¼ kz ¼ 1:5W m1 K 1 ; ar ¼ 5:1 106 K 1 ; c ¼ 420 J kg1 K 1 ; Yr ¼ 61:0 109 N m2 ; c13 ¼ 75:4 109 N m2 ; c33 ¼ 111:0 109 N m2 ; b1 ¼ 1:52 106 N K 1 m2 ; b3 ¼ 1:53 106 N K 1 m2 ; g3 ¼ 7:35 109 C2 N 1 m2 ; d1 ¼ 171 1012 C N 1 ; e1 ¼ 5:4C m2 ; e2 ¼ 15:8C m2 ; p3 ¼ 452 106 C K 1 m2 ; where ar is the radial coefficient of linear thermal expansion, c is the specific heat, Yr is Young’s modulus, d1 is a piezoelectric coefficient relating radial strain and transverse electric field intensity, and the values of ar ; b1 ; b3 and p3 are assumed. In order to investigate the effect of the piezoelectricity on the stresses, numerical calculations also have been performed for a material with thermoelastic properties identical to those of PZT-5A, but without the piezoelectric effect (e1 ¼ e3 ¼ p3 ¼ 0). For convenience in the presentation of numerical results, we introduce the following dimensionless quantities: me0 t me0 t0 j T rik jd1 jU ; t0 ¼ ; Bi ¼ hb; ce0 ¼ ; T ¼ ; rik ¼ ; ; U¼ b me0 b Tc ar Tc b b ar Yr T c pffiffiffiffiffiffiffiffiffiffiffiffi where me0 ¼ c33 =q is the velocity of stress wave propagation in the non-piezoelectric material. Biot’s number is taken to be Bi ¼ 10, the inertia parameter is taken to be ce0 ¼ 0:1, and two values of the dimensionless relaxation time are considered, namely t0 ¼ 0:05 and 0:5. Time histories of the temperature T, displacement uz , stresses rrr ; rhh ; rzz , and electric potential z z¼ ; b
t¼
10
F. Ashida and T. R. Tauchert
difference V ¼ Uðz; tÞ Uð0; tÞ at various locations z, corresponding to relaxation time t0 ¼ 0:05, are plotted in Figs. 1–5. Figures 6–10 show time histories of the same response quantities in the case of t0 ¼ 0:5. Solid-line curves represent results for the PZT-5A plate, whereas the broken-line curves in Figs. 2–4 and 7–9 denote corresponding results for the nonpiezoelectric plate. These figures illustrate that the relaxation time has a significant influence on the displacement, stresses and electric potential difference, but has little effect on the temperature. It is found that the thermal wave propagates more rapidly than the stress waves when t0 ¼ 0:05; conversely the stress waves travel more rapidly than the thermal wave when t0 ¼ 0:5. Comparing Figs. 3 and 4 with 8 and 9, it is seen that the peaks of the stresses rrr ; rhh ; rzz in the case of t0 ¼ 0:05 are much sharper and of far greater value than those for t0 ¼ 0:5.
1
z– = 1
0.8 z– = 0.75 0.6 z– = 0.5
T
–
0.4 –z = 0.25 0.2 –z = 0 0 –t = 0.05 0 –0.2 0
1
2
t–
3
4
5
4
5
Fig. 1. Time histories of temperature T in the case of t0 ¼ 0:05
t–0 = 0.05
1
–z = 1 –z = 0.75
u– z
0.5
–z = 0.5
0
–z = 0
–0.5
PZT – 5A plate Non-piezoelectric plate –1 0
1
2
3
t–
Fig. 2. Time histories of transverse displacement uz in the case of t0 ¼ 0:05
11
Thermally-induced wave propagation 3
–t = 0.05 0 z– = 0.5
2
1
– – srr , sqq
z– = 0 0 z– = 1
–1
–2
–3
PZT – 5A plate Non – piezoelectric plate
–4 0
1
2
–t
3
8
4
5
Fig. 3. Time histories of radial stress rrr and circumferential stress rhh in the case of t0 ¼ 0:05
– t0 = 0.5
PZT – 5A plate Non – piezoelectric plate
6
–z = 0.5
4
– s zz
2
0 –2
–z = 0.1
–4
–6 0
1
2
– t
3
4
5
Fig. 4. Time histories of transverse stress rzz in the case of t0 ¼ 0:05
Figures 3 and 8 indicate that the maximum values of the absolute radial and circumferential stresses in the PZT-5A plate are smaller than those in the non-piezoelectric plate. However, in the case of t0 ¼ 0:05, it is seen from Fig. 3 that the maximum values of tensile radial and circumferential stresses are larger in the PZT-5A plate. For t0 ¼ 0:5, we see from Fig. 8 that whereas tensile radial and circumferential stresses occur in the PZT-5A plate, these stress components remain compressive in the non-piezoelectric plate. Figures 4 and 9 show that the maximum values of the tensile and compressive transverse zz in the PZT-5A plate are greater than those in the non-piezoelectric plate for both stress r t0 ¼ 0:05 and t0 ¼ 0:5.
12
F. Ashida and T. R. Tauchert 0.1
–z = 0
–z = 0.25
0
– V
–0.1
–0.2 –z = 0.5
–0.3
–z = 0.75 –0.4 –z = 1
–t = 0.05 0 –0.5 0
1
2
3
– t
4
5
Fig. 5. Time histories of electric potential difference V in the case of t0 ¼ 0:05
–z = 1
1
0.8
–z = 0.75
0.6
– T
–z = 0.5 0.4 –z = 0.25 0.2 –z = 0 0 – t0 = 0.5 –0.2 0
1
2
t–
3
4
5
Fig. 6. Time histories of temperature T in the case of t0 ¼ 0:5
5 Concluding remarks Thermally-induced wave propagation in a thin piezoelectric plate has been analyzed, considering a relaxation time of heat flux. An exact solution to this problem was obtained by employing the Laplace transform technique. Numerical calculations were carried out for circular piezoelectric and non-piezoelectric plates subject to sudden heating. The numerical results show that, for the situations considered, the relaxation time has a significant influence on the displacement, stresses and electric potential difference across the plate, but has little effect on the temperature field. It was also found that the elastic and electric fields in the piezoelectric plate differ significantly from those in the non-piezoelectric plate.
13
Thermally-induced wave propagation
–t = 0.5 0
1
–z = 1 –z = 0.75
–u z
0.5
–z = 0.5
0
z– = 0 –0.5 PZT – 5A plate Non – piezoelectric plate –1 0
1
2
–t
3
5
4
Fig. 7. Time histories of transverse displacement uz in the case of t0 ¼ 0:5
1 PZT – 5A plate Non – piezoelectric plate 0.5
–t = 0.5 0
–z = 0.5
–z = 0
s– rr , s– qq
0
–0.5 –z = 1 –1
–1.5 0
1
2
– t
3
4
5
Fig. 8. Time histories of radial stress rrr and circumferential stress rhh in the case of t0 ¼ 0:5
Appendix Coefficients introduced in (49)–(50) and (51)–(53):
1 c2m 1 3jc2m am c2m 1 jc2m ; F2m ¼ ; F1m ¼ þ 2 þ 2t0 k2 m2e t20 me t0 2t0 k2 m2e t20 m2e t0 cm b cm b F10m ¼ ðF1m F9m F2m F8m Þ sin ; F11m ¼ ðF1m F8m þ F2m F9m Þ sin ; k k
14
F. Ashida and T. R. Tauchert 1 t– 0 = 0.5 –z = 0.5
s–z z
0.5
0
–z = 0.1
–0.5
PZT – 5A plate Non – piezoelectric plate –1 0
1
2
– t
3
5
4
Fig. 9. Time histories of transverse stress rzz in the case of t0 ¼ 0:5
0.1 –z = 0 0 –z = 0.25 –0.1
– V
–z = 0.5 –0.2 –z = 0.75
–0.3
–0.4 –z = 1
–t = 0.5 0 –0.5 0
F12m ¼
1
2
– t
3
4
5
Fig. 10. Time histories of electric potential difference V in the case of t0 ¼ 0:5
2n1 jc2m Am ; 2 þ F2 Þ am ðF1m 2m
F17m ¼ ð1 a2m ÞF11m þ 2am F10m ; F19m ¼ ð1 a2m ÞF2m þ 2am F1m ;
F18m ¼ ð1 a2m ÞF10m 2am F11m ; F20m ¼ ð1 a2m ÞF1m 2am F2m ;
in which n
o sinh 2tb0 me cos 2tam0 mbe ; cosh 2tb0 me sin 2tam0 mbe : fF8m ; F9m g ¼ sinh 2 2tb0 me cos 2 2tam0 mbe þ cosh 2 2tb0 me sin 2 2tam0 mbe
15
Thermally-induced wave propagation
Also K1m ¼
1 c2m 1 3jc2m ; þ 2t0 k2 m2e t20 m2e t0
K2m ¼
K10m ¼ ðK1m K9m K2m K8m Þ sin K12m ¼
cm b ; k
em c2m 1 jc2m ; þ 2t0 k2 m2e t20 m2e t0
K11m ¼ ðK1m K8m K2m K9m Þ sin
cm b ; k
2n1 jc2m Am ; 2 þ K2 Þ em ðK1m 2m
K17m ¼ ð1 þ e2m ÞK11m 2em K10m ; K19m ¼ ð1 þ e2m ÞK2m þ 2em K1m ;
K18m ¼ ð1 þ e2m ÞK10m 2em K11m ; K20m ¼ ð1 þ e2m ÞK1m þ 2em K2m ;
in which n
o sinh 2tb0 me cosh 2tem0 mbe ; cosh 2tb0 me sinh 2tem0 mbe : fK8m ; K9m g ¼ sinh 2 2tb0 me cosh 2 2tem0 mbe cosh 2 2tb0 me sinh 2 2tem0 mbe Furthermore, G12j ¼
2G7j ; G28j þ G29j
G13j ¼ G2j G8j G1j G9j ;
G14j ¼ G1j G8j þ G2j G9j ;
where G1j G3j G4j G5j
gj b qj b gj b qj b ¼ sinh cos ; G2j ¼ cosh sin ; k k k k gj b qj b gj b qj b cos qj sinh sin þ khG1j ; ¼ gj cosh k k k k gj b qj b gj b qj b cos þ qj sinh sin þ khG2j ; ¼ gj cosh k k k k m2e x2j t0 1 3 3 ; ¼ ; G ¼ m x 6j e j jk2 jk2 m2e
G7j ¼ ð1Þj
n1 khme Tc ; b
G8j ¼ G3j G5j G4j G6j ;
G9j ¼ G3j G6j þ G4j G5j ;
in which rffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi me x j 4 1 þ ðt0 me xj Þ2 ð cos hj sin hj Þ; ð cos hj þ sin hj Þ ; ðgi ; qj Þ ¼ 2j and hj ¼
1 tan1 ðt0 me xj Þ: 2
References [1] Ha, S. K., Keilers, C., Chang, F.-K.: Finite element analysis of composite structures containing distributed piezoceramic sensors and actuators. AIAA J. 30, 772–780 (1992).
16
F. Ashida and T. R. Tauchert: Thermally-induced wave propagation
[2] Abramovich, H., Livshits, A.: Dynamic behavior of cross-ply laminated beams with piezoelectric layers. Compos. Struct. 25, 371–379 (1993). [3] Yang, S. M., Chiu, J. W.: Smart structures – vibration of composites with piezoelectric materials. Compos. Struct. 25, 381–386 (1993). [4] Tzou, H. S., Ye, R.: Analysis of piezoelastic structures with laminated piezoelectric triangle shell elements. AIAA J. 34, 110–115 (1996). [5] Bruch, J. C., Sloss, J. M., Adali, S., Sadek, I. S.: Modified bang-bang piezoelectric control of vibrating beams. Smart Mater. Struct. 8, 647–653 (1999). [6] Tylikowski, A.: Control of circular plate vibrations via piezoelectric actuators shunted with a capacitive circuit. Thin-walled Struct. 39, 83–94 (2001). [7] Rao, S. S., Sunar, M.: Analysis of distributed thermopiezoelectric sensors and actuators in advanced intelligent structures. AIAA J. 31, 1280–1286 (1993). [8] Tzou, H. S., Ye, R.: Piezothermoelasticity and precision control of piezoelectric systems: theory and finite element analysis. J. Vibr. Acoust. 116, 489–495 (1994). [9] Lee, H.-J., Saravanos, D. A.: Coupled layerwise analysis of thermopiezoelectric composite beams. AIAA J. 34, 1231–1237 (1996). [10] Chandrashekhara, K., Tenneti, R.: Thermally induced vibration suppression of laminated plates with piezoelectric sensors and actuators. Smart Mater. Struct. 4, 281–290 (1995). [11] Tauchert, T. R., Ashida, F.: Control of thermally-induced structural vibrations via piezoelectric pulses. Proc. IUTAM Symp. on Dynamics of Advanced Materials and Smart Structures, Yonezawa, Japan, Kluwer Academic Publishers (in press). [12] Wang, X. D., Huang, G. L.: Wave propagation in electromechanical structures: induced by surface-bonded piezoelectric actuators. J. Intell. Mater. Sys. Struct. 12, 105–115 (2001). [13] Chandrasekharaiah, D. S.: Thermoelasticity with second sound: A review. Appl. Mech. Rev. 39, 355–376 (1986). [14] Ignaczak, J.: Generalized thermoelasticity and its applications. In: Thermal Stress III (Hetnarski, R. B., ed.), pp. 279–354. Amsterdam: North-Holland 1989. [15] Ignaczak, J.: Domain of influence results in generalized thermoelasticity – a survey. Appl. Mech. Rev. 44, 375–382 (1991). [16] Lord, H. W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967). Authors’ addresses: F. Ashida, Shimane University, Matsue, Schimane, Japan; T. R. Tauchert, Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506-0503, USA (E-mail: [email protected])
Acta Mechanica Printed in Austria
Thermally-induced wave propagation in a piezoelectric plate F. Ashida, Shimane, Japan, and T. R. Tauchert, Lexington, Kentucky Received October 3, 2002 Published online: February 10, 2003 Ó Springer-Verlag 2003
Summary. The Laplace transform technique is utilized to obtain the exact solution for thermally-induced wave propagation in a circular piezoelectric plate. The temperature is assumed to satisfy the one-dimensional heat conduction equation that includes a relaxation time. Displacement and electric potential functions are introduced in order to solve the equations of motion and electrodynamics. Numerical results obtained for both a PZT-5A plate and a non-piezoelectric plate illustrate the effects of relaxation time and piezoelasticity on displacement and stress histories.
1 Introduction Numerous papers have dealt with dynamic problems involving electromechanical structures. Most of the investigations have treated vibration problems, such as detection and/or control of harmonic vibrations in various host structures by means of attached piezoelectric sensors and/ or actuators. References [1]–[6], for example, consider piezo control of structural vibration under isothermal conditions, while [7]–[11] consider vibration suppression of structures exposed to thermal environments. Although the literature on harmonic vibrations of piezoelectric structures is plentiful, little attention has been given to wave propagation in piezoelectric materials. One study in this area is that of Wang and Huang [12], who analyzed an isothermal problem of wave propagation in an electro-elastic structure when the host structure is excited by means of piezoelectric actuators. It is expected that piezoceramic thin single crystal films of nano-order thickness will be put to practical use in the near future. When thermal loads act on such thin films, effects of both a relaxation time of heat flux and inertia effects in the thermoelastic field may be significant and should not be neglected. Inclusion of a relaxation time is necessary in order to eliminate the paradox of infinite velocity of a thermal wave, as is predicted by classical heat conduction theory. The importance of considering a relaxation time in the analysis of wave propagation in thermoelastic bodies is well documented (see Chandrasekharaiah [13] and Ignaczak [14], [15]). The present paper deals with the dynamic thermoelastic problem of a thin piezoelectric plate of crystal class 6mm. A relaxation time of heat flux, namely that attributed to Lord and Shulman [16], is taken into account. One boundary surface of the plate is exposed to a sudden uniform ambient temperature, whereas the other boundary surface is kept at zero temperature. An exact solution to this problem is obtained by employing the Laplace transform technique.
2
F. Ashida and T. R. Tauchert
Numerical calculations have been carried out for a thin PZT-5A plate, and the results are illustrated graphically.
2 Problem statement Consider a thin circular piezoelectric plate having material symmetry properties of crystal class 6 mm. The radius and thickness of the plate are a and b, respectively. It is assumed that initially the temperature throughout the plate is zero, and that the plate is free of deformation, stress, electric displacement and electric potential.
2.1 Temperature field Suppose that at time t ¼ 0 the top surface of the plate (z ¼ b) is exposed to a uniform ambient temperature Tc . The lower surface (z ¼ 0) is maintained at zero temperature, and the cylindrical edge (r ¼ a) is thermally insulated. In this case, the thermal initial and boundary conditions are expressed as T ¼ 0;
T;t ¼ 0 at t ¼ 0;
ð1Þ
T;z þ hT ¼ hTc
on
z ¼ b;
ð2Þ
T¼0
on
z ¼ 0;
ð3Þ
T;r ¼ 0
on
r ¼ a;
ð4Þ
where T denotes temperature, and h is the relative surface heat transfer coefficient. The temperature is assumed to satisfy the one-dimensional heat conduction equation which includes a relaxation time t0 , namely 1 k2 T;zz ¼ ðT;t þ t0 T;tt Þ: j
ð5Þ
Here k2 ¼ kz =kr in which kz and kr are coefficients of thermal conductivity, and j represents thermal diffusivity.
2.2 Elastic and electric field In determining the elastic and electric fields induced by the suddenly applied ambient temperature Tc , it is assumed that the cylindrical edge of the plate is smoothly constrained against radial deformation and free of electric charge. Then ur ¼ 0;
rrz ¼ 0;
ð6Þ
Er ¼ 0;
Dr ¼ 0;
ð7Þ
while all other response quantities are functions only of the spatial coordinate z and time t; in particular rrr ¼ rhh ¼ c13 uz;z e1 Ez b1 T;
ð8Þ
rzz ¼ c33 uz;z e3 Ez b3 T;
ð9Þ
Dz ¼ e3 uz;z þ g3 Ez þ p3 T;
ð10Þ
3
Thermally-induced wave propagation
where ui are elastic displacements, rik are stresses, Ei are electric field intensities, Di are electric displacements, cik are elastic moduli, ei are piezoelectric coefficients, bi are stress-temperature coefficients, g3 is a dielectric permittivity, and p3 is a pyroelectric constant. In this case, the elastic field is governed by the equation of motion rzz;z ¼ quz;tt ;
ð11Þ
where q denotes mass density, and the electric field satisfies the equation of electrodynamics Dz;zt ¼ 0:
ð12Þ
The upper and lower faces of the plate are considered to be free of both traction and electric charge. In this case the initial and boundary conditions are expressed as uz ¼ uz ;t ¼ 0 at t ¼ 0;
ð13Þ
Ez ¼ 0
at t ¼ 0;
ð14Þ
rzz ¼ 0
on z ¼ 0; b;
ð15Þ
Dz ¼ 0
on z ¼ 0; b:
ð16Þ
3 Analysis 3.1 Temperature field To determine the temperature field that satisfies the heat conduction Eq. (5) and initial conditions (1), we apply the Laplace transform Z 1 T ¼ Tept dt: ð17Þ 0
Equations (5), (2) and (3) become p k2 T;zz ¼ ð1 þ t0 pÞT ; j hTc T;z þ hT ¼ on z ¼ b; p T ¼ 0
on z ¼ 0:
ð18Þ ð19Þ ð20Þ
The solution to (18) that satisfies boundary condition (20) is taken to be rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pð1 þ t0 pÞ z : T ¼ A sinh j k
ð21Þ
Substitution of (21) into (19) gives k hT qffiffiffiffiffiffiffiffiffiffiffiffiffiffi c qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; A ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1þt0 pÞ pð1þt0 pÞ b pð1þt0 pÞ b p cosh j j j k þ k h sinh k
ð22Þ
in which case (21) becomes qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pð1þt0 pÞ z k hTc sinh j k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : T ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1þt0 pÞ pð1þt0 pÞ b pð1þt0 pÞ b p cosh j j j k þ k h sinh k
ð23Þ
4
F. Ashida and T. R. Tauchert
Inversion of T is performed using the method of residues, with the result that
1 X z am t 1 am t t þ Am H a2m cos sin T ¼ A0 þ e 2t0 b 2t0 am 2t0 m¼1 em t 1 em t c z þ sin m ; þ Hðe2m Þ sinh 2t0 em 2t0 k
ð24Þ
where HðxÞ is the Heaviside unit step function and A0 ¼
am ¼
hbTc ; 1 þ hb
Am ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4jt0 c2m 1;
2 k hTc n o ; cm ð1 þ hbÞ cos cmk b cmk b sin cmk b em ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4jt0 c2m ;
ð25Þ
ð26Þ
and cm are the roots of the equation cm b cm b þ k h sin ¼ 0: cm cos k k
ð27Þ
3.2 Elastic and electric fields In order to solve the equations of motion (11) and electrodynamics (12), we introduce a displacement potential X and an electric potential U defined by the relations uz ¼ X;z ;
ð28Þ
Ez ¼ U;z :
ð29Þ
Substitution of (28) and (29) into (8)–(10) gives the following expressions for the stresses and electric displacement: rrr ¼ rhh ¼ c13 X;zz þ e1 U;z b1 T;
ð30Þ
rzz ¼ c33 X;zz þ e3 U;z b3 T;
ð31Þ
Dz ¼ e3 X;zz g3 U;z þ p3 T:
ð32Þ
By introducing (30)–(32) into (11) and (12), we obtain the equations governing the displacement and electric potentials, namely X;zz
1 X;tt ¼ n1 T; m2e
ð33Þ
U;z ¼ dX;zz þ n2 T; in which 1 e2 c33 þ 3 ; m2e ¼ q g3
ð34Þ
n1 ¼
g 3 b 3 e 3 p3 ; g3 c33 þ e23
n2 ¼
p3 ; g3
d¼
e3 : g3
ð35Þ
In this case the electric displacement Dz is identically zero and thus the boundary condition (16) has been satisfied. The aforementioned initial conditions, namely (13) and (14), imply that X ¼ X;t ¼ U ¼ 0
at t ¼ 0:
ð36Þ
Next applying the Laplace transform to (33) and (34) under the initial conditions (36) gives
Thermally-induced wave propagation
X;zz
p2 X ¼ n1 T ; m2e
5
ð37Þ
U;z ¼ dX;zz þ n2 T :
ð38Þ
Substituting (23) for T into (37) and (38) and then integrating gives rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pð1 þ t0 pÞ z pz þ E sinh X ¼ C sinh ; j k me rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pð1 þ t0 pÞ z pz U ¼ D cosh þ F cosh ; j k me
ð39Þ
ð40Þ
where C ¼ n1 khTc
p2
1 t0 1 þ jk2 jk2 m2e
(rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !)# pð1 þ t0 pÞ pð1 þ t0 pÞ b pð1 þ t0 pÞ b cosh þ k h sinh j j k j k " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1 dð1 þ t0 pÞ 1 t0 1 pð1 þ t0 pÞ 1 t0 1 2 þ n2 k p hTc p p þ þ D¼ j j jk2 m2e jk2 m2e jk2 jk2 (rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !)# pð1 þ t0 pÞ pð1 þ t0 pÞ b pð1 þ t0 pÞ b cosh þ k h sinh ; j j k j k F¼
ð41Þ
d p E: me
By combining (11) and (28), the axial stress is expressed as rzz ¼ qX;tt :
ð42Þ
Taking the Laplace transform of this equation gives rzz ¼ qp2 X :
ð43Þ
Substituting (39) into (43) then yields ( rzz
2
¼ qp
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ) pð1 þ t0 pÞ z pz þ E sinh : C sinh j k me
ð44Þ
Applying boundary condition (15), written in the transform domain as rzz ¼ 0
on z ¼ 0; b;
ð45Þ
leads to the result sinh
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð1þt0 pÞ b j k
E¼ sinh
pb me
C:
ð46Þ
6
F. Ashida and T. R. Tauchert
Substitution of (41) and (46) for the coefficients C; D; E and F into (39) and (40) gives ( rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ) pb pð1 þ t0 pÞ z pð1 þ t0 pÞ b pz sinh sinh X ¼ n1 khTc sinh sinh me j k j k me
" 1 t0 1 pb þ p sinh p2 2 2 2 me me jk jk (rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !)# pð1 þ t0 pÞ pð1 þ t0 pÞ b pð1 þ t0 pÞ b cosh þ kh sinh ; j j k j k
ð47Þ
and "
(rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! pð1 þ t0 pÞ pb pð1 þ t0 pÞ z sinh U ¼ hTc n1 d cosh j me j k
kp sinh me
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! )
pð1 þ t0 pÞ b pz 1 t0 1 pb cosh p sinh p þ j k me me jk2 jk2 m2e
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !,rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# pð1 þ t0 pÞ z pð1 þ t0 pÞ þ n2 k cosh j k j 2
," (rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !)# pð1 þ t0 pÞ pð1 þ t0 pÞ b pð1 þ t0 pÞ b cosh þ kh sinh : p j j k j k
ð48Þ
Inversion of X and U by the method of residues leads to the results X¼
n 1 A0 3 ðz b2 zÞ 6b þe
2tt
0
1 X
Hða2m ÞF12m
m¼1
c z am t am t F1m sin sin m F2m cos 2t0 2t0 k
am t am t z am z þ F11m sin sinh cos þ F10m cos 2t0 2t0 2t0 me 2t0 me
am t am t z am z F10m sin cosh sin F11m cos 2t0 2t0 2t0 me 2t0 me þ
Hðe2m ÞK12m
c z em t em t K1m sinh sin m K2m cosh 2t0 2t0 k
em t em t z em z þ K11m sinh sinh cosh þ K10m cosh 2t0 2t0 2t0 me 2t0 me
em t em t z em z K11m cosh þ K10m sinh cosh sinh 2t0 2t0 2t0 me 2t0 me
1 X j¼1
G12j G13j cos ðme xj tÞ þ G14j sin ðme xj tÞ sin ðxj zÞ;
ð49Þ
Thermally-induced wave propagation
7
and U¼
A0 n 1 d n2 b2 t t ð3z2 b2 Þ þ z2 b2 þ hb z2 þ n2 jk2 t0 1 þ e t0 t0 b 6 2ð1 þ hbÞ 3
1 c z X dc F12m am t am t t e 2t0 F1m sin F2m cos cos m Hða2m Þ m 2t0 2t0 k k m¼1
dF12m am t am t þ ðam F11m F10m Þ cos ðF11m þ am F10m Þ sin 2t0 2t0 2t0 me z am z am t cos þ ðF11m þ am F10m Þ cos cosh 2t0 me 2t0 me 2t0 ) am t z am z sinh sin þ ðam F11m F10m Þ sin 2t0 2t0 me 2t0 me
c z n kAm am t 1 am t þ cos m þ 2 cos sin 2t0 am 2t0 k cm
c z dc K12m em t em t þ Hðe2m Þ m K1m sinh K2m cosh cos m 2t0 2t0 k k
dK12m em t em t ðK11m em K10m Þ sinh þ ðem K11m K10m Þ cosh 2t0 2t0 2t0 me z em z em t cosh þ ðK11m em K10m Þ cosh cosh 2t0 me 2t0 me 2t0
em t z em z sinh sinh ðem K11m K10m Þ sinh 2t0 2t0 me 2t0 me
n2 kAm em t 1 em t cm z þ cos þ cosh sinh 2t0 em 2t0 k cm d
1 X
xj G12j G13j cos ðme xj tÞ þ G14j sin ðme xj tÞ cos ðxj zÞ;
ð50Þ
j¼1
where xj ¼ jp=b, and the coefficients Fim ; Kim ði ¼ 1; 2; 10; 11; 12Þ and Gij ði ¼ 12; 13; 14Þ are defined in the Appendix. The transverse elastic displacement, found by substituting (49) into (28), becomes uz ¼
n 1 A0 ð3z2 b2 Þ 6b þe
2tt
0
1 X
Hða2m ÞF12m
m¼1
c z cm am t am t F2m cos F1m sin cos m 2t0 2t0 k k
1 am t am t þ ðF11m þ am F10m Þ sin ðF10m am F11m Þ cos þ 2t0 me 2t0 2t0 cosh
z 2t0 me
cos
am z 2t0 me
8
F. Ashida and T. R. Tauchert
1 am t am t ðF10m am F11m Þ sin ðF11m þ am F10m Þ cos 2t0 me 2t0 2t0
z 2t0 me
sinh
cos
sin
am z 2t0 me
þ Hðe2m ÞK12m
cm em t em t K2m cosh K1m sinh 2t0 2t0 k
c z 1 em t em t m þ ðem K10m K11m Þ sinh ðem K11m K10m Þ cosh k 2t0 me 2t0 2t0
z em z 1 em t cosh ðem K10m K11m Þ cosh cosh þ 2t0 me 2t0 me 2t0 me 2t0
em t z em z þ ðem K11m K10m Þ sinh sinh sinh 2t0 2t0 me 2t0 me
1 X
xj G12j G13j cos ðme xj tÞ þ G14j sin ðme xj tÞ cos ðxj zÞ:
ð51Þ
j¼1
Substitution of (24), (49) and (50) into (30) and (42) yields the following expressions for the elastic stresses:
ðc13 þ e1 dÞn1 rrr ¼ rhh ¼ b1 þ e1 n2 A0 z b e
2tt
0
1 X m¼1
ðc13 þ e1 dÞc2m F12m am t am t F F Hða2m Þ cos sin 2m 1m 2 2t0 2t0 k
c z am t 1 am t þ sin m cos sin 2t0 am 2t0 k
þðb1 e1 n2 ÞAm
ðc13 þ e1 dÞF12m ð2t0 me Þ2
F18m
am t am t z am z þ F17m sin sinh cos cos 2t0 2t0 2t0 me 2t0 me
am t am t z am z F18m sin cosh sin F17m cos 2t0 2t0 2t0 me 2t0 me
Hðe2m Þ
ðc13 þ e1 dÞc2m K12m em t em t K1m sinh K2m cosh 2t0 2t0 k2
þðb1 e1 n2 ÞAm
c z em t 1 em t þ sin m cosh sinh 2t0 em 2t0 k
ðc13 þ e1 dÞK12m ð2t0 me Þ2
em t em t z em z þ K17m sinh sinh cosh K18m cosh 2t0 2t0 2t0 me 2t0 me
em t em t z em z K17m cosh þ K18m sinh cosh sinh 2t0 2t0 2t0 me 2t0 me þ ðc13 þ e1 dÞ
1 X j¼1
x2j G12j G13j cos ðme xj tÞ þ G14j sin ðme xj tÞ sin ðxj zÞ;
ð52Þ
Thermally-induced wave propagation
9
1 q 2tt X am t am t cm z 2 0 F sin e Hða ÞF F cos sin 12m 19m 20m m 2t0 2t0 c 4t20 m¼1
am t am t z am z þ F17m sin sinh cos þ F18m cos 2t0 2t0 2t0 me 2t0 me
am t am t z am z F18m sin cosh sin F17m cos 2t0 2t0 2t0 me 2t0 me
em t em t c z þ Hðe2m ÞK12m K19m cosh K20m sinh sin m 2t0 2t0 k
em t em t z em z þ K18m cosh þ k17m sinh sinh cosh 2t0 2t0 2t0 me 2t0 me
em t em t z em z þ K18m sinh cosh sinh K17m cosh 2t0 2t0 2t0 me 2t0 me
rzz ¼
þ qm2e
1 X
x2j G12j G13j cos ðme xj tÞ þ G14j sin ðme xj tÞ sin ðxj zÞ;
ð53Þ
j¼1
where the coefficients Fim ; Kim ði ¼ 17; 18; 19; 20Þ are defined in the Appendix.
4 Numerical results Numerical calculations have been carried out for a PZT-5A plate. The material constants are taken to be: q ¼ 7750 kg m3 ; kr ¼ kz ¼ 1:5W m1 K 1 ; ar ¼ 5:1 106 K 1 ; c ¼ 420 J kg1 K 1 ; Yr ¼ 61:0 109 N m2 ; c13 ¼ 75:4 109 N m2 ; c33 ¼ 111:0 109 N m2 ; b1 ¼ 1:52 106 N K 1 m2 ; b3 ¼ 1:53 106 N K 1 m2 ; g3 ¼ 7:35 109 C2 N 1 m2 ; d1 ¼ 171 1012 C N 1 ; e1 ¼ 5:4C m2 ; e2 ¼ 15:8C m2 ; p3 ¼ 452 106 C K 1 m2 ; where ar is the radial coefficient of linear thermal expansion, c is the specific heat, Yr is Young’s modulus, d1 is a piezoelectric coefficient relating radial strain and transverse electric field intensity, and the values of ar ; b1 ; b3 and p3 are assumed. In order to investigate the effect of the piezoelectricity on the stresses, numerical calculations also have been performed for a material with thermoelastic properties identical to those of PZT-5A, but without the piezoelectric effect (e1 ¼ e3 ¼ p3 ¼ 0). For convenience in the presentation of numerical results, we introduce the following dimensionless quantities: me0 t me0 t0 j T rik jd1 jU ; t0 ¼ ; Bi ¼ hb; ce0 ¼ ; T ¼ ; rik ¼ ; ; U¼ b me0 b Tc ar Tc b b ar Yr T c pffiffiffiffiffiffiffiffiffiffiffiffi where me0 ¼ c33 =q is the velocity of stress wave propagation in the non-piezoelectric material. Biot’s number is taken to be Bi ¼ 10, the inertia parameter is taken to be ce0 ¼ 0:1, and two values of the dimensionless relaxation time are considered, namely t0 ¼ 0:05 and 0:5. Time histories of the temperature T, displacement uz , stresses rrr ; rhh ; rzz , and electric potential z z¼ ; b
t¼
10
F. Ashida and T. R. Tauchert
difference V ¼ Uðz; tÞ Uð0; tÞ at various locations z, corresponding to relaxation time t0 ¼ 0:05, are plotted in Figs. 1–5. Figures 6–10 show time histories of the same response quantities in the case of t0 ¼ 0:5. Solid-line curves represent results for the PZT-5A plate, whereas the broken-line curves in Figs. 2–4 and 7–9 denote corresponding results for the nonpiezoelectric plate. These figures illustrate that the relaxation time has a significant influence on the displacement, stresses and electric potential difference, but has little effect on the temperature. It is found that the thermal wave propagates more rapidly than the stress waves when t0 ¼ 0:05; conversely the stress waves travel more rapidly than the thermal wave when t0 ¼ 0:5. Comparing Figs. 3 and 4 with 8 and 9, it is seen that the peaks of the stresses rrr ; rhh ; rzz in the case of t0 ¼ 0:05 are much sharper and of far greater value than those for t0 ¼ 0:5.
1
z– = 1
0.8 z– = 0.75 0.6 z– = 0.5
T
–
0.4 –z = 0.25 0.2 –z = 0 0 –t = 0.05 0 –0.2 0
1
2
t–
3
4
5
4
5
Fig. 1. Time histories of temperature T in the case of t0 ¼ 0:05
t–0 = 0.05
1
–z = 1 –z = 0.75
u– z
0.5
–z = 0.5
0
–z = 0
–0.5
PZT – 5A plate Non-piezoelectric plate –1 0
1
2
3
t–
Fig. 2. Time histories of transverse displacement uz in the case of t0 ¼ 0:05
11
Thermally-induced wave propagation 3
–t = 0.05 0 z– = 0.5
2
1
– – srr , sqq
z– = 0 0 z– = 1
–1
–2
–3
PZT – 5A plate Non – piezoelectric plate
–4 0
1
2
–t
3
8
4
5
Fig. 3. Time histories of radial stress rrr and circumferential stress rhh in the case of t0 ¼ 0:05
– t0 = 0.5
PZT – 5A plate Non – piezoelectric plate
6
–z = 0.5
4
– s zz
2
0 –2
–z = 0.1
–4
–6 0
1
2
– t
3
4
5
Fig. 4. Time histories of transverse stress rzz in the case of t0 ¼ 0:05
Figures 3 and 8 indicate that the maximum values of the absolute radial and circumferential stresses in the PZT-5A plate are smaller than those in the non-piezoelectric plate. However, in the case of t0 ¼ 0:05, it is seen from Fig. 3 that the maximum values of tensile radial and circumferential stresses are larger in the PZT-5A plate. For t0 ¼ 0:5, we see from Fig. 8 that whereas tensile radial and circumferential stresses occur in the PZT-5A plate, these stress components remain compressive in the non-piezoelectric plate. Figures 4 and 9 show that the maximum values of the tensile and compressive transverse zz in the PZT-5A plate are greater than those in the non-piezoelectric plate for both stress r t0 ¼ 0:05 and t0 ¼ 0:5.
12
F. Ashida and T. R. Tauchert 0.1
–z = 0
–z = 0.25
0
– V
–0.1
–0.2 –z = 0.5
–0.3
–z = 0.75 –0.4 –z = 1
–t = 0.05 0 –0.5 0
1
2
3
– t
4
5
Fig. 5. Time histories of electric potential difference V in the case of t0 ¼ 0:05
–z = 1
1
0.8
–z = 0.75
0.6
– T
–z = 0.5 0.4 –z = 0.25 0.2 –z = 0 0 – t0 = 0.5 –0.2 0
1
2
t–
3
4
5
Fig. 6. Time histories of temperature T in the case of t0 ¼ 0:5
5 Concluding remarks Thermally-induced wave propagation in a thin piezoelectric plate has been analyzed, considering a relaxation time of heat flux. An exact solution to this problem was obtained by employing the Laplace transform technique. Numerical calculations were carried out for circular piezoelectric and non-piezoelectric plates subject to sudden heating. The numerical results show that, for the situations considered, the relaxation time has a significant influence on the displacement, stresses and electric potential difference across the plate, but has little effect on the temperature field. It was also found that the elastic and electric fields in the piezoelectric plate differ significantly from those in the non-piezoelectric plate.
13
Thermally-induced wave propagation
–t = 0.5 0
1
–z = 1 –z = 0.75
–u z
0.5
–z = 0.5
0
z– = 0 –0.5 PZT – 5A plate Non – piezoelectric plate –1 0
1
2
–t
3
5
4
Fig. 7. Time histories of transverse displacement uz in the case of t0 ¼ 0:5
1 PZT – 5A plate Non – piezoelectric plate 0.5
–t = 0.5 0
–z = 0.5
–z = 0
s– rr , s– qq
0
–0.5 –z = 1 –1
–1.5 0
1
2
– t
3
4
5
Fig. 8. Time histories of radial stress rrr and circumferential stress rhh in the case of t0 ¼ 0:5
Appendix Coefficients introduced in (49)–(50) and (51)–(53):
1 c2m 1 3jc2m am c2m 1 jc2m ; F2m ¼ ; F1m ¼ þ 2 þ 2t0 k2 m2e t20 me t0 2t0 k2 m2e t20 m2e t0 cm b cm b F10m ¼ ðF1m F9m F2m F8m Þ sin ; F11m ¼ ðF1m F8m þ F2m F9m Þ sin ; k k
14
F. Ashida and T. R. Tauchert 1 t– 0 = 0.5 –z = 0.5
s–z z
0.5
0
–z = 0.1
–0.5
PZT – 5A plate Non – piezoelectric plate –1 0
1
2
– t
3
5
4
Fig. 9. Time histories of transverse stress rzz in the case of t0 ¼ 0:5
0.1 –z = 0 0 –z = 0.25 –0.1
– V
–z = 0.5 –0.2 –z = 0.75
–0.3
–0.4 –z = 1
–t = 0.5 0 –0.5 0
F12m ¼
1
2
– t
3
4
5
Fig. 10. Time histories of electric potential difference V in the case of t0 ¼ 0:5
2n1 jc2m Am ; 2 þ F2 Þ am ðF1m 2m
F17m ¼ ð1 a2m ÞF11m þ 2am F10m ; F19m ¼ ð1 a2m ÞF2m þ 2am F1m ;
F18m ¼ ð1 a2m ÞF10m 2am F11m ; F20m ¼ ð1 a2m ÞF1m 2am F2m ;
in which n
o sinh 2tb0 me cos 2tam0 mbe ; cosh 2tb0 me sin 2tam0 mbe : fF8m ; F9m g ¼ sinh 2 2tb0 me cos 2 2tam0 mbe þ cosh 2 2tb0 me sin 2 2tam0 mbe
15
Thermally-induced wave propagation
Also K1m ¼
1 c2m 1 3jc2m ; þ 2t0 k2 m2e t20 m2e t0
K2m ¼
K10m ¼ ðK1m K9m K2m K8m Þ sin K12m ¼
cm b ; k
em c2m 1 jc2m ; þ 2t0 k2 m2e t20 m2e t0
K11m ¼ ðK1m K8m K2m K9m Þ sin
cm b ; k
2n1 jc2m Am ; 2 þ K2 Þ em ðK1m 2m
K17m ¼ ð1 þ e2m ÞK11m 2em K10m ; K19m ¼ ð1 þ e2m ÞK2m þ 2em K1m ;
K18m ¼ ð1 þ e2m ÞK10m 2em K11m ; K20m ¼ ð1 þ e2m ÞK1m þ 2em K2m ;
in which n
o sinh 2tb0 me cosh 2tem0 mbe ; cosh 2tb0 me sinh 2tem0 mbe : fK8m ; K9m g ¼ sinh 2 2tb0 me cosh 2 2tem0 mbe cosh 2 2tb0 me sinh 2 2tem0 mbe Furthermore, G12j ¼
2G7j ; G28j þ G29j
G13j ¼ G2j G8j G1j G9j ;
G14j ¼ G1j G8j þ G2j G9j ;
where G1j G3j G4j G5j
gj b qj b gj b qj b ¼ sinh cos ; G2j ¼ cosh sin ; k k k k gj b qj b gj b qj b cos qj sinh sin þ khG1j ; ¼ gj cosh k k k k gj b qj b gj b qj b cos þ qj sinh sin þ khG2j ; ¼ gj cosh k k k k m2e x2j t0 1 3 3 ; ¼ ; G ¼ m x 6j e j jk2 jk2 m2e
G7j ¼ ð1Þj
n1 khme Tc ; b
G8j ¼ G3j G5j G4j G6j ;
G9j ¼ G3j G6j þ G4j G5j ;
in which rffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi me x j 4 1 þ ðt0 me xj Þ2 ð cos hj sin hj Þ; ð cos hj þ sin hj Þ ; ðgi ; qj Þ ¼ 2j and hj ¼
1 tan1 ðt0 me xj Þ: 2
References [1] Ha, S. K., Keilers, C., Chang, F.-K.: Finite element analysis of composite structures containing distributed piezoceramic sensors and actuators. AIAA J. 30, 772–780 (1992).
16
F. Ashida and T. R. Tauchert: Thermally-induced wave propagation
[2] Abramovich, H., Livshits, A.: Dynamic behavior of cross-ply laminated beams with piezoelectric layers. Compos. Struct. 25, 371–379 (1993). [3] Yang, S. M., Chiu, J. W.: Smart structures – vibration of composites with piezoelectric materials. Compos. Struct. 25, 381–386 (1993). [4] Tzou, H. S., Ye, R.: Analysis of piezoelastic structures with laminated piezoelectric triangle shell elements. AIAA J. 34, 110–115 (1996). [5] Bruch, J. C., Sloss, J. M., Adali, S., Sadek, I. S.: Modified bang-bang piezoelectric control of vibrating beams. Smart Mater. Struct. 8, 647–653 (1999). [6] Tylikowski, A.: Control of circular plate vibrations via piezoelectric actuators shunted with a capacitive circuit. Thin-walled Struct. 39, 83–94 (2001). [7] Rao, S. S., Sunar, M.: Analysis of distributed thermopiezoelectric sensors and actuators in advanced intelligent structures. AIAA J. 31, 1280–1286 (1993). [8] Tzou, H. S., Ye, R.: Piezothermoelasticity and precision control of piezoelectric systems: theory and finite element analysis. J. Vibr. Acoust. 116, 489–495 (1994). [9] Lee, H.-J., Saravanos, D. A.: Coupled layerwise analysis of thermopiezoelectric composite beams. AIAA J. 34, 1231–1237 (1996). [10] Chandrashekhara, K., Tenneti, R.: Thermally induced vibration suppression of laminated plates with piezoelectric sensors and actuators. Smart Mater. Struct. 4, 281–290 (1995). [11] Tauchert, T. R., Ashida, F.: Control of thermally-induced structural vibrations via piezoelectric pulses. Proc. IUTAM Symp. on Dynamics of Advanced Materials and Smart Structures, Yonezawa, Japan, Kluwer Academic Publishers (in press). [12] Wang, X. D., Huang, G. L.: Wave propagation in electromechanical structures: induced by surface-bonded piezoelectric actuators. J. Intell. Mater. Sys. Struct. 12, 105–115 (2001). [13] Chandrasekharaiah, D. S.: Thermoelasticity with second sound: A review. Appl. Mech. Rev. 39, 355–376 (1986). [14] Ignaczak, J.: Generalized thermoelasticity and its applications. In: Thermal Stress III (Hetnarski, R. B., ed.), pp. 279–354. Amsterdam: North-Holland 1989. [15] Ignaczak, J.: Domain of influence results in generalized thermoelasticity – a survey. Appl. Mech. Rev. 44, 375–382 (1991). [16] Lord, H. W., Shulman, Y.: A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15, 299–309 (1967). Authors’ addresses: F. Ashida, Shimane University, Matsue, Schimane, Japan; T. R. Tauchert, Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506-0503, USA (E-mail: [email protected])