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УДК 539.3
Аналитическое решение задачи напряжений в магнито-пьезоэлектрическом термоупругом материале под действием вращения
K.S. Al-Basyouni1, E. Ghandourah2, B. Dakhel1
Университет короля Абдулазиза, Джидда, 21589, Саудовская Аравия
В рамках линейной теории термоупругости изучено распространение волн в однородном поперечно-изотропном магнито-термо-пьезоэлектрическом материале в форме цилиндра под действием вращения. Для разделения уравнений движения вводятся функции смещения, электрического потенциала, магнитного потенциала, электростатического смещения и магнитной индукции, а также граничные условия. Получены характеристические уравнения в замкнутой форме и изолированные математические условия, с помощью которых изучено влияние вращения, фазовой скорости, зависимостей коэффициента затухания и относительного сдвига частоты на поведение дисперсионных кривых. Результаты исследования могут быть использованы при разработке датчиков вращения и других пьезоэлектрических приборов. Численные результаты для физических величин представлены графически.
Ключевые слова: вращение, электромагнитные волны, поперечно-изотропный, пьезоэлектрический, термоупругий
DOI 10.24411/1683-805X-2019-15013
An analytical solution for the problem of stresses in magneto-piezoelectric thermoelastic material under the influence of rotation
K.S. Al-Basyouni1, E. Ghandourah2,3, and B. Dakhel1
1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia 2 Department of Nuclear Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, 21589, Saudi Arabia 3 GRC Department, Jeddah Community College, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
The present paper deals with the study of the wave propagation in a homogeneous transversely isotropic magneto-thermo-piezoelec-tric cylindrical material, under the influence of rotation, in the context of the linear theory of thermoelasticity. The displacement, electric potential, magnetic potential, electrostatic displacement, and magnetic induction functions are introduced to uncouple the equations of motion, with boundary conditions. After deriving secular equations in closed form and isolated mathematical conditions, the effect of rotation, phase velocity, attenuation coefficient profiles and relative frequency shift on dispersion curves is studied. The study is very helpful in the development of rotation sensors and other piezoelectric devices. The numerical results of the physical quantities have been illustrated graphically.
Keywords: rotation, electromagnetic waves, transversely isotropic, piezoelectric, thermoelastic
1. Introduction
Propagation of waves in piezoelectric materials is an active research subject for engineers and scientists for several decades because of the application in piezoelectric transducers, resonators, filters, resonators, actuators, and other devices such as microelectromechanical systems [14]. The interaction between the magnetic and thermal fields plays a vital role in geophysics for understanding the effect of Earth's magnetic field on seismic waves. With the de-
velopment of active material systems, there is a significant interest in the coupling effects between the elastic, magnetic, and temperature for their application in sensing and actuation [5, 6]. Wang and Dai [7] presented magneto-ther-modynamic stresses and perturbation of the magnetic field vector in an orthotropic thermoelastic cylinder. Using the theory of magneto-thermoelastic, vibration of functionally graded multilayered orthotropic cylindrical panel under thermomechanical load was analyzed by Wang [8].
© Al-Basyouni K.S., Ghandourah E., Dakhel B., 2019
Sharma [9] studied the propagation of the plane harmonic thermoelastic wave in homogenous transversely isotropic, cubic crystals, and anisotropic materials in the context of generalized thermoelasticity. Sharma and Sidhu [10] investigated the three-dimensional vibration analysis of a transversely isotropic thermoelastic cylindrical panel. Mahmoud [11, 12] study the influence of rotation and generalized magneto-thermoelastic on Rayleigh waves in a granular medium under the effect of initial stress and gravity field. Chadwick [13] studied the propagation of plane harmonic waves in homogenous anisotropic heat conducting solids, an infinite magneto-electroelastic cylinder. Buchanan [14] demonstrated that the application of powerful numerical tools like finite element or boundary element methods to these problems are also becoming important. Chen [15, 16] has obtained vibration analysis of orthotropic cylindrical shells with free ends by the Rayleigh-Ritz method, point temperature solution for a penny-shaped crack in an infinite transversely isotropic thermopiezoelastic medium. Pon-nusamy [17] has obtained the frequency equation of free vibration of a generalized thermoelastic solid cylinder of the arbitrary cross-section by using the Fourier expansion collocation method. The theory of magneto-thermoelasticity has aroused many applications in many industrial appliances, particularly in nuclear devices, where a primary magnetic field exists. Abd-Alla and Mahmoud [18-21] discussed a magneto-thermoelastic problem in rotating nonho-mogeneous orthotropic hollow cylindrical under the hyperbolic heat conduction model. Mukhopodhyay and Roy-choudhury [22] discussed the magneto-thermoelastic interaction in an infinite isotropic elastic cylinder subjected to periodic loading. Recently, Hosseini and Hossein [23] developed the analytical solution for thermoelastic waves in the thick hollow cylinder based on Green-Naghdi model of coupled thermoelasticity. He studied the thermome-chanical behavior in the time domain using the fast Laplace inverse transform method.
Chitikireddy et al. [24] studied the transient thermal elastic waves in an anisotropic hollow cylinder due to localized heating. He adopted the generalized thermoelasticity theory proposed by Lord and Shulman and used the integral transform method in the frequency and wavenumber domain to get the modal waveforms. Mahmoud et al. [2527] studied the effect of the rotation on wave motion through a cylindrical bore in a micropolar porous cubic crystal. Sharma and Walia [28, 29] investigated the effect of rotation on Rayleigh waves in piezothermoelastic half-space, charge and stress-free piezothermoelastic plate in the context of the conventional coupled theory of piezothermoelas-ticity. They studied the wave characteristics, such as phase velocity and attenuation coefficient of the waves in cadmium selenide (CdSe) material.
In the present work, the one-dimensional wave propagation, under the influence of rotation, in a homogeneous
transversely isotropic magneto-piezothermoelastic cylindrical panel is discussed using the linear theory of elasticity. The displacement, electric potential, magnetic potential, electrostatic displacement, and magnetic induction functions are introduced to uncouple the equations of motion, with boundary conditions. The effect of rotation, phase velocity, attenuation coefficient profiles, and relative frequency shift on dispersion curves is studied. The computed nondimensional parameters are plotted in the form of dispersion curves.
2. Formulation of the problem
The coordinate system with (r, 6, z) of length L of the cylindrical material, under the influence of rotation, and the magnetic field, having an inner and outer radius a and b with thickness h, r is the radius of the material. The distortion of the cylindrical material is in the radial direction u. The cylindrical material is assumed to be homogeneous, transversely isotropic and linearly elastic. Young's modulus E and density p are in an undisturbed state. In the cylindrical coordinates (r, 6, z), the motion equations in the absence of anybody force for linear elastic mediums is given as
3T„ .1 , ____ d2--
- + - (Tr-T66 ) + p(Q xfi xu)r =p -T,
dr r dt
(1)
where ft = (0,0, Q), (ft x ft x u)r is the component of the centripetal acceleration in the radial direction r, due to the time-varying motion only. For transversely isotropic mediums, the Lord-Shulman generalized heat conduction equation is as follows:
ki(T- + r ) = pCJTt +ToTt ] +
+ To
dt+to at2
x (Pl(^rr + )).
(2)
The simplified Maxwell charge equilibrium equations for electric and magnetic fields are given below. The strain mechanical displacement relation is given as
dM 1 (3)
dr
= — u, r
where u are displacements along the radial direction, err, e66 are normal strain components.
Using the generalized Hooke's law, the stress-strain relations for transversely isotropic materials are given as
Trr = c11err + c12e66 - PlT,
T66 = c11err + c12e66 - P1T, where
p1 = (c11 + c12 ) y1 + c13y3- (5)
The electric displacement and magnetic induction are related regarding strain, electric field, and magnetic field in the following form:
Dr =-EnV,r -m$r, (6)
Br =-mnVr -M,r, (7)
where t„
are the stress components, err, eee are the
strain components, cn, c12 are the independent elastic constants, p, Cv are the mass density and specific heat at constant strain, >, Di and Bt are the electric potential, magnetic potential, electrostatic displacement, and magnetic induction, Ejk and Mjk are the dielectric and magnetic permeability coefficients, ekj, dkj and mjk are the piezoelectric, piezomagnetic and magnetoelectric material coefficients, Y1, k1 is the coefficient of linear thermal expansion and thermal conductivities along and perpendicular to the axis of symmetry. Substituting Eqs. (3)-(7) in Eqs. (1), (2) will give the following four displacement equations of motion:
d2u 1 ( du 1 ^ „ dT d2u
cii
dr r
2
c11
-e
11
-m.
11
dr 2
aV
dr 2
dr dr 2
r J ^ dr P dr2
d> d2> 1 ( a^ 3é . „ -m»-d + mn^ 1 = 0,
dr
dr
d2> 1 (
dr 2
a>
dr
dr
(8) (9) (10)
(
1ar + d 2t
r dr dr2
V y
= PCv
dT dt
2m \
- + Tn
d_T
dt2
+TA
( a2
du + 1 du dtdr r dt
\
+ T0TP
( a\
1 a 2u ^
+
dt2dr r dt2
(11)
3. The solution of the problem
Equations (8)-(11) are coupled with both odd ordered and even ordered derivatives of displacement, electric potential, magnetic potential and heat components with respect to one specific coordinate variable. The mechanical displacement is determined by Sharma [9] as given below:
u = ~ U, (12)
dr
where u(r, t) is the displacement potentials for symmetric modes of vibrations. To uncouple Eqs. (8)-(11), we can write four displacement components, temperature, and magnetic potential, which satisfy the stress-free boundary conditions as follows:
u(r, t) = U(r)eiat, V(r, t) = - Y(r)eiœt,
a
>(r, t ) = - O (r)eimt, T (r, t ) = -T (r )eiœt, a a
(13)
where m is the angular frequency of the cylindrical material. Introduce the following dimensionless quantities:
t=— o2 =£ml m = mu£n, p* = PL,
'o,
^ ' 1 rl
c11 e31d31
P = Tft, M1 =
M-11c11
e =-
ei ici
d31
11c11
e11
Substituting Eq. (13) and (14) into Eqs. (8)-(11), we obtain
v? -
+4
r
U + PT = 0,
e1V12T + m1V12O = 0,
m1V12T + M1v2o = 0,
(15)
(16) (17)
(
(
k[Vf -PC
(
52 ^
- + T0
dt 0 at2
t -
- TP
2 d ro2T0 -—-T dt
J J \2 \
dt2
V{U = 0,
(18)
1, V 2 1 3 + 32 where v1 = —— + —2.
r dr dr
We assume that the disturbance is time harmonic through the factor eiœt.
The solutions of Eqs. (15)-(18) are obtained as
U (r ) = Cs + C6 J0 (B1r ) + C7 J0 (B2 r ) +
+ C8 K^r ) + C9 J0 (B24 r ), T (r ) = Bs((C9K0(B6r) + C7 J0 (B7 r))Bg + (19)
+ C6 J0(B9r ) + C8 K0(B10 r )), T(r) = C1 + C2 ln r, O(r) = C3 + C4 ln r. Substituting Eqs. (19) into Eqs (13), we obtain U (r, t ) = (Cs + C6 J0 Br ) + C7 J0 (B2 r) + + C8 K>(B3r ) + C9 K0(B, r ))eiœt,
V(r, t) = - (C1 + C2 ln r)eiat, a
>(r, t) = - (C3 + C4 ln r )eiœt, a
T (r, t ) = -Bs ((C9 K0 (B6r ) + C7 J0 (B7 r ))B + a
+ C6 J,(B9r ) + C8 K0(Bwr ))eiœt. Substituting Eqs. (201) into Eq. (12) we obtain u(r, t) = (C6Jx(Br)B + C7Jl(B1r)B2 + + C8 KxBr )B23 + C9Kx(BAr )B,)eiœt,
V(r, t) = - (C1 + C2 ln r)eiat, a
>(r, t) = - (C3 + C4ln r)eiœt, a
T(r, t) = Bs ((C9K0 (B6r) + C7 J0 (B7r))B8 + + C6 J »Br ) + C8K>(B10r ))eiœt.
Now substituting Eqs. (21) into Eqs. (3)-(7), we get the stresses as
(20)
(21)
&rr = c11
( ( C6
V v
J¡¡(Br) +
J1(B1r)
(
+ C7
Jq( B2 r) +
+ C
K0( B3r) +
+ C9
Kq( B4r) +
J1B r)
B2r
2
K1(B3r) B3r
K1( B4r) B4r
Br
B +
B22 +
B32 +
B42
irot ,
e +
+ - (C12(C6 J^r) B + C7 J1( B2 r) B2 +
r
+ C8 J1( B3r) B3 + C9 J1( B4 r) B4)eirot) -- ppB5((C9K0(B7r) + C7J1(B8r))B5 + + (C6 J0 (B9r) + C8 K0 (B10 r)) Blb)dwt,
a66 = c12
(( C6
V
J0( B?) -
J1(B1r) B1r
B12 +
+ C7
J0(B2r) -
(
+ Cs
K0( B3r) -
(
- C9
K^B.r) +
J1(B2r) I B22 +
B2r 2
K^r) B32 -
B3r 3
K1( B4 r) B42
B4r
irot ,
e +
/ y
+ - (c^^r )B + C7 J1(B2r )B2 + r
+ C8 K1 (B3r)B3 + C9 K1 (B4r)B4 )eirot) -- P1B5 ((C9K0 (B7r) + C7 J0 (B8r))B8 + + (C6 J0 (B9 r) + C8 K0 (B10 r))B6)eirot,
Dr =-—(C2E11 + C4mn)eirot, ra
Br =-—(C2 mn + C4M-H )eirot, ra
where
B1 = K
B2=\ 2 2 ^
B3 = — i
2(A2 + ^/A")ln e ®
c11k1
2(A2 -yfA)lnero
c11k1 '
2(A2 e ro
B4 = — i
2 1
2'V
c11k1
2(A2 -yfA)lnero
c11k1
(22)
B5 =-■1 i((-T0 Inero + i)7A --ro2T0((Cvpcn +T0pj2)T0 -k1p)ln2e + + iro((CvpC11 + TjPj2 )T0 - 2k1p) ln e + + 2 TiPj2 + 2 C^pcn),
B6 =
1T0 ln e ro-1 i 2 0 2
VA -
-2ro\((Cvpc„ + T0pj2)T0 -k1p)ln2e + + iro( (CBpcn + T0p!2)T0 - 2k1p Iln e +
1-o2 1
+2 T0P12 + 2 Cvpcn
b7 = -1 i
2 V
2(-^A3 - A4)ln e ro
c11k1 '
B=
1
2V
2(-^A3 - A4)ln e ro
c11k1 '
B9 =-
1
- A4)ln
e ro
c11k1
B10 = -11
where
2
2(Ja3 - A4)ln e ro
c11k1
A = C2 lnero2p2t0c121 + 2Cv lneT0ro2p2t0p2c11 + + ln eT02ro2x2P4 - 2iCV ln e rop2T0c121 -
- 2Cv ln e ro2p2T0c11k1 - 4iCv ln eT0ropx0P'^c11 + + 2 lneT0ro2pT0P?k - 2i lneT02rox0P4 - Cv2p2c121 + + 2iCv lnerop2c11k1 - 2CvT0pP12k1 + lnero2p2k12 -
- 2ln eTT0ropP2kx - T02P4,
A2 = -Cv lneropx0c11 - lneT0roT0P22 + + icvpc11 + iToP2 - lneropk1,
A3 = ((Cvpcn + T0P?)2 Tg + 2k1p(-Cvpcn + TiP2 )T0 + + p2k2)ro2 ln2 e -2i((Cvpcn + TPP )2 T0 + + k1p(-Cvpc11 + T0P2)) lnero - Cvpc11 + T0P12,
A4 = ro(CvpT0c11 + T^2 + pk1) ln e + i(T0Pl2 + Cvpcn), roln e
A =
k1P1(-T0ln e ro + i)
4. Boundary conditions
In this section, the cylinder subjected to boundary conditions on the upper and lower surfaces in r = a, b. The nondimensional mechanical boundary conditions for a stress-free edge are given by Grr = 0, r = a, b.
The nondimensional boundary condition for thermally insulated or isothermal surfaces is Tr + hT = 0, where h is the surface heat transfer coefficient, here h ^ 0 corresponds to a a thermally insulated surface and h ^ ^ refers to an isothermal one.
The nondimensional magnetic and electric boundary condition is given as
Br,r = 0, Dr,r = 0.
5. Numerical results and discussion
In order to achieve the numerical computations, we regard the closed circular cylindrical material. For transversely isotropic materials, the elastic constants are taken from Buchanan [14]
c11 = 218x109 N/m2, c12 = 120x109 N/m2, m11 = 0.0074 x 10"9 Ns/VC, p = 7500 kg, = -200 x10-6, e11 = 0.4 x10"9 C/Vm, P1 = 0.621 x106 N Km, en = 10, ce = 260x 106 J kg"1 K-1, p = 5.3, a = 0.8, b = 1.4, h = b/a, k = 1.5, Cv = 420x106, T0 = 298. The corresponding plots of nondimensional phase velocity and attenuation coefficient with nondimensional wave number are given in Figs. 1, 2 and the nondimensional rotation rate has been taken as O = 0.5 in Figs. 1-3. The corresponding plots of relative frequency shift with thickness h are given in Fig. 3. The phase velocity profiles are plotted in Fig. 1 for charge-free (open circuit) surface boundary. It evident that the waves at long wavelength have substantially greater sensitivity to rotation than at short wavelengths. The effect of rotation on waves in the considered material is quite large in case of charge-free (open circuit)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 Wave number
<
o.ooo-|
0.0 0.5 1.0 1.5 2.0 2.5 3.0 Wave number
Fig. 2. Dispersion curves for attenuation coefficient in open circuit, stress free and thermally insulated with wave number for different cases: O = 0.0 (1), 0.5 (2)
boundary compared to electrically shorted (closed circuit) one. The wave speed of piezothermoelastic wave approaches to that of thermoelastic wave in the absence of rotation (dotted curves). Moreover, the phase speed in the presence of rotation (solid curves) also approximately coincides with piezothermoelastic wave speed as the wave number progresses towards large values. The effect of anisotropy is also quite pertinent from the plots. The phase velocity increases due to the rotation for the charge-free boundary and suffers a decrease in case of electroded one.
Figure 2 represent attenuation coefficient profiles of waves in case of charge-free as a boundary condition, under rotation and nonrotation environment. The attenuation profiles vary in a nonlinear manner at small wave numbers (long wavelengths) and linearly at high wave numbers (short wavelengths) in this case. The attenuation is increased due to the rotation for an open circuit (charge free) and surfers a decrease due to rotation in case of closed circuit (electroded). The magnitude of attenuation is many times more in case of electroded than for charge free one as can be noticed from the respective calculation in our work and previous work.
3
Fig. 1. Dispersion curves for phase velocity in open circuit, stress free and thermally insulated with wave number for different cases: O = 0.0 (1), 0.5 (2)
Fig. 3. Variation of relative frequency shift in open circuit, stress free and thermally insulated with thickness h in different cases: O = 0.0 (1), 0.5 (2)
Figure 3 represents variations of relative frequency shifts with thickness h in the case of open circuit. The relative frequency shift varies linearly with small increasing in thickness h in case of electroded surfaces and charge-free surfaces but nonlinearly for large values of thick. These rotation sensitivity characteristics are of interest for the development of rotation sensors, pyro- and piezoelectric devices for which frequency insensitivity to rotation is desired.
According to the results mentioned above, we can conclude that the analytical solutions, under the influence of the rotation, based on the normal mode medium to analyze the magneto-piezoelectric thermoelastic material has been developed and used. All physical quantities coincide with the boundary conditions. Finally, these results also provide engineers and designers with a useful benchmark for modeling and design solution. We can see that the analytical solutions, under the influence of the rotation and the magnetic field based on analysis of normal mode for heat elastic piezoelectric medium, has been developed and used. All physical quantities values converge to a minimal value with increasing distance.
6. Conclusion
The mathematical model of wave propagation under the effect of rotation, the heat and the magnetic field, homogeneous piezoelectric cylindrical material, as well as transversely isotropic with using linear thermal theory is studied. The linear stresses, displacement functions, function of magnetic effort, function of electrostatics and the function of magnetic induction have used in the equations of motion with the appropriate conditions. The corresponding relations of nondimensional phase velocity and attenuation coefficient with nondimensional wavenumber are given, and the corresponding relation of relative frequency shift with rotation are discussed. The results represented graphically to explain the physical aspects of the study.
Funding
This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant No. G-397-130-33. The authors, therefore, acknowledge with thanks DSR for technical and financial support.
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Received July 29, 2019, revised July 29, 2019, accepted August 26, 2019
CeedeHua 06 aemopax
Khalil Salem Al-Basyouni, Dr., King Abdulaziz University, Saudi Arabia, [email protected] E. Ghandourah, King Abdulaziz University, Saudi Arabia B. Dakhel, King Abdulaziz University, Saudi Arabia
