CC BY
10
Математика. Физика
УДК 539.3
DOI: 10.17277/vestnik.2020.02.pp.306-322
COUPLED THERMO-ELECTRO-MECHANICAL ANALYSIS OF HYBRID LAYERED CYLINDRICAL SHELLS WITH PIEZOELECTRIC SENSORS AND ACTUATORS
G. M. Kulikov, N. P. Merkusheva, M. G. Kulikov, S. V. Plotnikova
Laboratory of Intelligent Materials and Structures, gmkulikov@mail.ru;
Tambov State Technical University, Tambov, Russia
Keywords: layered piezoelectric shell; thermopiezoelectricity; 3D stress analysis; sampling surfaces method.
Abstract: This paper focuses on the implementation of the sampling surfaces (SaS) method for solving the three-dimensional coupled problems for layered piezoelectric shells subjected to thermal and electromechanical loading. The SaS method is based on choosing inside the nth layer In SaS parallel to the middle surface in order to introduce the temperatures, electric potentials and displacements of these surfaces as basic unknowns. Such choice of unknowns with the consequent use of the Lagrange polynomials of degree In -1 in assumed approximations of the temperature, electric potential, displacements and mechanical material properties through the thickness of the layer leads to an efficient thermopiezoelectric shell model. The inner SaS are located inside each layer at Chebyshev polynomial nodes that allows one to minimize uniformly the error due to the Lagrange interpolation of high order. As a result, the SaS method can be applied efficiently for obtaining the analytical solutions for hybrid layered cylindrical shells with piezoelectric sensors and actuators, which asymptotically approach the exact solutions of thermopiezoelectricity as the number of SaS In tends to infinity.
Introduction
Three-dimensional (3D) analysis of layered piezoelectric plates and shells under thermal loading has received considerable attention during past twenty years (see, e.g. [1, 2]). There are at least five approaches to 3D exact solutions of thermoelectroelasticity for piezoelectric plates and shells, namely, the Pagano approach [3 - 7], the state space approach [8 - 12], the power series expansion approach [13 - 15], the asymptotic expansion approach [16], and the sampling surfaces (SaS) approach [17 - 19].
In this paper, the SaS approach is utilized for the first time for the coupled thermoelectroelastic stress analysis of hybrid layered composite shells with piezoelectric sensors and actuators. According to the SaS approach [20], we choose arbitrarily located surfaces inside the nth layer parallel to the middle surface of a shell
Q(n)1, Q(n)2,..., dn)I" to introduce temperatures T(n)1, T(n)2,..., T(n)I« , electric
potentials 9(n)2,..., q{n)I" and displacement vectors u(n)1, u(n)2,..., u(n)I" of
these surfaces as basic shell variables, where In is the total number of SaS of the nth layer (In > 3 ). Such choice of temperatures, electric potentials and displacements with the consequent use of the Lagrange polynomials of degree In -1 in the assumed distributions of the temperature, electric potential, displacements and mechanical properties through the thickness of the layer allows the presentation of governing equations of the SaS shell formulation in a very compact form.
It should be noted that the SaS shell formulation with equally spaced SaS [21 - 23] does not work properly with the Lagrange polynomials of high degree because of Runge's phenomenon [24]. This phenomenon yields the wild oscillation at the edges of the interval when the user deals with some specific functions similar to the shell metric functions. If the number of equispaced nodes is increased then the oscillations become even larger. However, the use of the Chebyshev polynomial nodes inside the shell body [20, 25, 26] can help to improve significantly the behavior of the Lagrange polynomials of high degree because such a choice permits one to minimize uniformly the error due to the Lagrange interpolation. This fact gives an opportunity to obtain the stresses with a prescribed accuracy using the sufficiently large number of SaS. It means in turn that the solutions based on the SaS concept asymptotically approach the 3D exact solutions of thermoelectroelasticity as In ^ & .
Description of displacement and strain fields
We consider a layered shell of the thickness h. Let the middle surface Q be described by orthogonal curvilinear coordinates 81 and 62, which are referred to the lines of principal curvatures of its surface. The thickness coordinate 93 is oriented in the normal direction. Introduce the following notations: Aa (81,62) are the coefficients of the first fundamental form; ka (81,62) are the principal curvatures of the middle
surface; ca = 1 + ka93 are the components of the shifter tensor; (0b62) are the
components of the shifter tensor at SaS of the nth layer Q(n)in (Fig. 1) defined as
can>n = Ca (63n)in ) = 1 + ka63n)in , (1)
where 63n)in are the transverse coordinates of SaS given by
03n)1 =03n-1], 03n)In =03n],
e(«)m« = i(0[«-1] +0[«]) - Ih{n)
2 4 3 3 ' 2
( 2mn- 3 ^
n---
2(In - 2)
(2)
where eE"-1] and 9" are the transverse coordinates of layer interfaces and
q[«] ; h(n) = 9[3"] -e3"-1] is the thickness of the nth layer. It is worth noting that the
transverse coordinates of inner SaS e3")m" coincide with the coordinates of the
Chebyshev polynomial nodes. This fact has a great meaning for the convergence of the SaS method [20].
Fig. 1. Geometry of the layered shell
Here, the index n identifies the belonging of any quantity to the nth layer and runs from 1 to N, where N is the number of layers; the index mn identifies the belonging of any quantity to the inner SaS of the nth layer and runs from 2 to In -1; the indices in, jn , kn describe all SaS of the nth layer and run from 1 to In; Latin tensorial indices i, j,k, l range from 1 to 3; Greek indices a, p range from 1 to 2.
We start now with the first two assumptions of the proposed layered piezoelectric shell formulation. The displacement and strain fields are distributed through the thickness of the nth layer [20] as follows:
(n) = £L(n)'nu(n)'n , 03n-1] < 03 < 03n],
in
j = £L(n)inepn, 03n-1] < 03 < 03n],
(3)
(4)
where u(n)ln (0b 02) and ej)ln (0j, 02) are the displacements and strains of SaS of the nth layer Q(n)in ; L(n)i« (03) are the Lagrange polynomials of degree In -1 given by
u(n)'n = Ui (03nVn),
eifn = eij (03n)ln),
L(n)ln = n
03 -03n) jn
jn *in 03
(n)in
-0
(n) jn
(5)
(6)
(7)
The strains of SaS of the nth layer in terms of displacements of SaS are expressed as
2e(n )in = 1 l(n )in + 1 l(n )in
2 P = c(n)in Aap + (n)in Apa '
cp ua
n
2p(«>« = 1 i(n)in + a(")in (n)in =R(« )i„ (8)
a3 = (n)in A3a +Pa , e33 = P3 ' (8)
ca
where are the strain parameters of SaS of the nth layer [20]; p(n)in = Mi3(e3n)in )
are the values of the derivative of displacements with respect to the thickness coordinate on SaS defined as
4n)n = A- uaa + B4n),n + ka«3n)in, = -r ian - B^n
Aa Aa
^ = -Mna,n - kauan)in for p*a, (9)
Aa
p(n)in = £M(n)■ (03n)in )u(n)■ , (10)
Jn
where M(n)Jn = L(nJn are the derivatives of Lagrange polynomials; their values on SaS are calculated as
1 0(n)in -0(n)kn
M(n)jn (03n)in ) = ( ) . 1 ( ). n 0(3) ■ 03 )k for jn * in,
03n) Jn -03n)inknin^ Jn 03n) Jn -03n)kn n
M(n)in (03n)in ) = - £M(n)Jn (03n)in ). (11)
Jn ^ ln
It is seen from formula (10) that the key functions p(n )in of the layered shell formulation are represented as a linear combination of displacements of SaS of the nth
layer ui(n) Jn .
Description of electric field
Next, we introduce the third and fourth assumptions of the proposed layered thermopiezoelectric shell formulation. The electric potential and the electric field are distributed through the thickness of the nth layer [18] as follows:
9(n) = £ L(n)'n 9(n)in , 03n-1] < 03 < 03n], (12)
.n
E(n) =£L(n)'nE(n)'n, 03n-1] <03 <03n], (13)
in
where 9(n)i" (0!,02) are the electric potentials of SaS of the nth layer; E.n)n (01;02) are the components of the electric field at SaS of the nth layer defined as
9(n)in = 9(03n)in), (14)
E(n)'n = E, (03n )in). (15)
The electric field on SaS of the nth layer in terms of electric potentials of SaS is given by
Ean)in =-—^r 9(a)i'n, (16)
A c(n)1n
E(in)ln = -£M(n) jn (03n)ln )9(n) jn. (17)
jn
As can be seen from formula (17), the normal components of the electric field on SaS of the nth layer E:(n)ln are represented as a linear combination of electric potentials
of SaS of the same layer ip(n)j" .
Description of temperature field
The following step consists in a choice of the suitable approximation of the temperature and temperature gradient through the shell thickness. The temperature and temperature gradient are distributed through the thickness of the nth layer [18] as follows:
T(n) = £L(n)lnT(n)ln, 03n-1] < 03 < 03n], (18)
in
r(n) =£L(n)lnr(n)ln, 03n-1] <03 <03n], (19)
in
where T(n)i" (0b 02) are the temperatures of SaS of the nth layer; r(")in (01; 02) are the components of the temperature gradient on SaS of the nth layer defined as
T (n)ln = T (03n)ln), (20)
r(n)ln = ri (03n)ln). (21)
The components of the temperature gradient on SaS of the nth layer in terms of temperatures of SaS [18] are expressed as
ran)ln =-l—— T^n, (22)
a A c(n)ln 'a
r3n)ln = £M(n)jn (03n)ln )T(n)jn. (23)
jn
It is seen from formula (23) that the normal components of the temperature gradient at SaS of the nth layer r3n)ln are represented as a linear combination of temperatures of SaS of the same
layer T(n)jn . Constitutive equations As constitutive equations, we accept Fourier heat conduction equations
q(n) = -kjn)r(n), 03n-1] < 03 < 03n], (24)
where q(n) are the components of the heat flux of the nth layer; kj) are the thermal conductivities. Here and below, the summation on repeated Latin indices is implied.
We introduce the next assumption of the thermal shell formulation. Let us assume that the thermal conductivity coefficients are distributed through the thickness of the nth layer as follows:
j = £ L(n)lnkJn>'n, 03n-1] < 03 < 03n] (25)
in
that is extensively utilized in this paper, where kj)ln = k\"")i0"")ln) are the values of the
thermal conductivity tensor on SaS of the nth layer.
For simplicity, we consider the case of linear piezoelectric materials. Therefore, the constitutive equations are expressed as
j = jekn) -ekjEn* -T^n)®(n), D^ = 4Mn) +^(kn) Ekn) + r/n)0(n),
n(n) =Ykn)ekn) + rkn ]Ekn) +X(n)®(n), (26)
where a(/n) are the stresses of the nth layer; D(n) are the electric displacements; n(n) is the entropy density; ©(n) = T(n) -T0 is the temperature rise; T0 is the reference temperature; Cj) are the elastic constants; eknj are the piezoelectric constants; y" are the thermal stress coefficients; e^ are the dielectric constants; rf-"^ are the pyroelectric constants; x(n) is the entropy-temperature coefficient given by
X(n) = P(n)4n)/ T0, (27)
where p(n) and c,n) are the mass density and the specific heat per unit mass.
Finally, we introduce the last assumption of the SaS thermopiezoelectric shell formulation. Let the material constants be distributed through the thickness of the nth layer as accepted throughout this paper
s(n) =£L(n)lnS(n)in, 03n-1] <03 <03n], (28)
in
W(n) = [C(n) e(n) Y(n) e(n) r(n) p(n) c(")] " = [Cijkl, eijk , Yij , eij , ri , P , Cv ],
where S(n)in = S(n)(03n)in) are the values of material constants on SaS of the nth layer.
Analytical solution for layered composite cylindrical shell
In this section, we study a layered composite cylindrical shell with embedded piezoelectric layers subjected to thermal and electro-mechanical loads. The boundary conditions for the simply supported cylindrical shell with electrically grounded edges maintained at the reference temperature are written as
©(«) = = CTl(1} = 4n) = u3n) = 0 at 0j = 0 and 0j = L, (29)
where 0j is the longitudinal coordinate; L is the length of the shell. To satisfy the boundary conditions (29), we search the analytical solution by a method of the double Fourier series expansion
TOTO p.
©(n )'n = £ £©№ sin Äcos se2, (30)
r=1s=0 L
TOTO p.
p(n)in = £ £cp(.n)'n sin^Llcosse2, (31)
r=1s=0 L
TOTO
u[n)in = ££ uWn cos^cosse2, (32)
r=1s=0 L
TOTO
u(2n)'n = ££u2nrS'n sin -rn^ism se2, (33)
r=1s=0 L
TOTO
(n)in ^¡r^r (n)in ■ rne1 u\ ' n = £ £ u n --
3 = 11"^ sin-^coss62, (34)
r=1s=0 L
where 62 is the circumferential coordinate; r, s are the wave numbers. The external electromechanical loads are also expanded in double Fourier series.
Substituting formula (30) in a variational equation of the heat conduction theory [18], we obtain the systems of linear algebraic equations in terms of temperatures
®("sof order K, where K = £nIn - N +1. Therefore, the temperatures of SaS can
be found using the method of Gaussian elimination. Substituting then (31) - (34) in a variational equation of the thermopiezoelectric shell theory [18], one obtains the
systems of linear algebraic equations in terms of $s)ln , ^ , M2nS n and u3nl'n
of order 4K, in which the temperatures of SaS ©^^n are known. These linear systems are solved again through the method of Gaussian elimination.
The described algorithm was performed with the Symbolic Math Toolbox, which incorporates symbolic computations into the numeric environment of MATLAB. This permits the obtaining of analytical solutions for layered composite cylindrical shells with piezoelectric layers in the framework of the SaS thermoelectroelastic shell formulation, which asymptotically approach the 3D exact solutions of thermopiezoelectricity as the number of SAS of the nth layer In tends to infinity.
Numerical examples
We consider a three-layer symmetric cross-ply cylindrical shell with the stacking sequence [90/0/90] composed of the graphite-epoxy composite and covered with two piezoelectric PVDF layers on its bottom and top surfaces. Thus, the hybrid five-layer cylindrical shell [PVDF/90/0/90/PVDF] with ply thicknesses [3h0/8h0/8h0/8h0/3h0]
is studied, where h0 = h /30. The material properties of the PVDF polarized in the thickness direction are taken to be [14]:
E1 = E2 = E3 = 2 GPa, v12 = v23 = v31 = 1/3,
au = a22 = a33 = 120x10"61/K, kn = k22 = k33 = 0.24 W/mK, d3n = 3x10-12m/V, d322 = 23x10-12m/V, d333 =-30x10-12m/V, eu=e22 = 3.078x10-UF/m, e33= 3.141x10-UF/m, r3 =-27x10-6C/m2K,
where Ei are the elastic moduli; vij- are the Poisson's ratios; aij- are the thermal coefficients of expansion; dijk are the piezoelectric moduli. The material properties of the graphite-epoxy composite are chosen as follows [15]:
El = 172.5GPa, ET = 6.9GPa, GLT = 3.45GPa, GTT = 1.38GPa, vLT = vTT = 0.25,
aL = 0.57x 10-6 1/K, aT = 35.6x10-6 1/K, kL = 36.42 W/mK, kT = 0.96W/mK,
eL = 3.095 x10-11 F/m, eT = 2.653x10-11 F/m, where L and T denote the fiber and transverse directions. To evaluate the entropy, we accept p = 1780 kg/m3, cv = 1400 J/kgK and p = 1800 kg/m3, cv = 900 J/kgK for the PVDF and graphite-epoxy, respectively.
The shell is subjected to sinusoidally distributed temperature loading on the top surface, whereas the bottom surface is maintained at the reference temperature. The bottom and top surfaces are electroded and grounded. Therefore, the boundary conditions can be written as
0+ = ©0 sin ^L1, 9+ = °+3 = ct+3 = CT+3 = a
0" = 9"=CT-3 =0-3 =0-3 = 0, (35)
where 00 = 1K and T0 = 293K. The geometric parameters of the shell are chosen to be R = 1m and L = 4m, where R is the radius of the middle surface. It is assumed that the interfaces between the piezoelectric layers and the substrate are electroded and grounded.
To compare the derived results with the 3D exact solution of thermoelectroelasticity [15], we introduce dimensionless variables at crucial points as follows:
© = ©(L /2, z) / ©0, 9 = 103 dr 9(L /2, z) / har©0, U1 = 100u1(0,z)/Rar©0, u3 = 10u3(L/2,z)/Rar©0,
q3 = 100hq3(L /2, z)/ kr©0, n= 10-3 n( L /2, z)/ Era^©0, 013 = 105013(0,z)/Erar©0, O33 = 10So33(L/2,z)/Erar©0, where z = 03/h is the dimensionless thickness coordinate; S = R /h is the slenderness
ratio; Er = 6.9 GPa,ar = 35.6x10-6 1/K, kr = 36.42 W/mKanddr = 30x10-12 m/V
are the representative moduli of the shell.
Figures 2, 3 display the distributions of the temperature, electric potential, displacements, heat flux, entropy and stresses through the thickness of the hybrid layered cylindrical shell for different values of the slenderness ratio employing seven SaS for each layer. A comparison with the 3D exact solution [15] is also presented. These results demonstrate convincingly the high potential of the developed SaS formulation because the boundary conditions on bottom and top surfaces of the shell for transverse stresses and the continuity conditions for the heat flux and transverse stresses at interfaces are satisfied exactly.
Next, we study a simply supported metal-ceramic cylindrical shell covered with the graphite-epoxy layer and PVDF at the bottom. Therefore, we deal here with a hybrid three-layer shell [PVDF/Graphite-Epoxy/Metal-Ceramic] with ply thicknesses [0.1h/0.1h/0.8h]. The fibers of the graphite-epoxy composite are oriented in the axial
0,25
0.25
0 0.25 0.5 0.75 0
0.5 \ i I1 0.5
0.25 - .v Д lot 100 0.25
0 - \ 0
-0.25 - >/ -0.25
-0.5 ij -0.5
1
100 \f/s= 2
if i
-18 -15 -12
-1.5 0 1.5
Fig. 2. Through-thickness distributions of the temperature, electric potential and displacements for a hybrid five-layer cylindrical shell: SaS formulation for seven SaS inside each layer and 3D exact solution (o) [15]
Fig. 3. Through-thickness distributions of the heat flux, entropy and stresses for a hybrid five-layer cylindrical shell: SaS formulation for seven SaS inside each layer
direction. The material properties of the graphite-epoxy and PVDF are presented in a previous example. The metal-ceramic shell is made of the two-phase composite.
The metal phase is aluminum with Em = 7 x
km = 233W/mK, pm = 2707kg/m3 and cm = 896J/kgK; the thermal barrier is a SiC
ceramic with Ec = 38x1010Pa, vc = 0.17, ac = 4.3x10-61/K, kc = 65W/mK,
pc = 3100kg/m3 and cc = 670J/kgK. It is assumed that the material properties are varied through the thickness according to the rule of mixtures:
E = EmVm + EcVc, v = vmVm + vcVc, a = amVm + «-0^
k = kmVm + kcVc, Pc = Pm cmVm + Pc ccVc, (36)
where Vm and Vc are the volume fractions of metal and ceramic phases defined as
Vm = 1 - Vc, Vc =[( z + 0.3)/0.8]y , z e[-0.3,0.5], (37)
where y is the material gradient index; z = 93/h is the dimensionless thickness coordinate.
Here, we study a cylindrical shell subjected to sinusoidally distributed temperature loading on the top surface and consider the following boundary conditions:
©+ = ©0 sin —cos 202, 9+ = ct+3 = ct+3 = ct+3 = 0,
©- = 0, D- = ct-3 = CT-3 = CT-3 = 0, (38)
where 0 0 = 1K and T0 = 293K. The interface between PVDF and graphite-epoxy is electroded and grounded.
To analyze the obtained results efficiently, we introduce dimensionless variables at crucial points as functions of the dimensionless thickness coordinate z = 93 /h as
© = ©(L/2,0, z)/©0, 9 = dr9(L/2,0, z)/har ©0, u1 = 10-4u1 (0,0,z)/Rar©0, U3 = 10-5u3(L/2,0,z)/Rar©0, au = 10-5 ctu(L /2,0, z)/ Erar©0, ct22 = 10-5 ct22(L /2,0, z)/ Erar©0, a12 = 10-4Sct12(0,n/4,z)/Erar©0, CT13 = 10-3SCT13(0,0,z)/Erar©0,
a23 = 10-3 Sct23(L /2, n /4, z)/ Erar©0, ct33 = 10-3 Sa33(L /2,0, z)/ Erar©0, S = R / h,
where Er and ar are the representative moduli taken from a previous example. The geometric parameters of the shell are chosen as R = 1m and L = 4m .
Figures 4, 5 show the through-thickness distributions of the temperature, electric potential, displacements and stresses for different values of the slenderness ratio S and material gradient index y = 2 using 13 SaS for each layer. These results demonstrate again the high potential of the developed SaS shell formulation, since the boundary conditions on the bottom and top surfaces and the continuity conditions at the interfaces for transverse stresses are satisfied properly.
Fig. 4. Through-thickness distributions of the temperature, electric potential and displacements for a hybrid three-layer cylindrical shell: SaS formulation for Y = 2 and 13 SaS inside each layer
Fig. 5. Through-thickness distributions of stresses for a hybrid three-layer cylindrical shell: SaS formulation for y = 2 and 13 SaS inside each layer (continued p. 317)
Fig. 5. Through-thickness distributions of stresses for a hybrid three-layer cylindrical shell: SaS formulation for y = 2 and 13 SaS inside each layer
Conclusions
A robust SaS formulation for the coupled steady-state thermal stress analysis of layered piezoelectric shells has been proposed. It is based on a concept of SaS located at Chebyshev polynomial nodes throughout the layers and interfaces as well. As a result, the developed SaS formulation makes it possible to obtain the Ritz solutions for hybrid layered cylindrical shells with piezoelectric sensors and actuators with a prescribed accuracy, which can asymptotically approach the 3D exact solutions of thermopiezoelectricity as the number of SaS goes to infinity.
This work was supported by the Russian Science Foundation (Grant No. 18-1900092).
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Связанный термоэлектромеханический анализ гибридных слоистых
цилиндрических оболочек с пьезоэлектрическими сенсорами
и актуаторами
Г. М. Куликов, Н. П. Меркушева, М. Г. Куликов, С. В. Плотникова
Научно-исследовательская лаборатория «Механика интеллектуальных материалов и конструкций», gmkulikov@mail.ru;
ФГБОУ ВО «ТГТУ», г. Тамбов, Россия
Ключевые слова: слоистая пьезоэлектрическая оболочка, термопьезоэлектричество, трехмерный анализ напряжений, метод отсчетных поверхностей.
Аннотация: Данная статья посвящена реализации метода отсчетных поверхностей (SaS метод) для решения трехмерных связанных задач для слоистых пьезоэлектрических оболочек, подверженных температурным и электромеханическим воздействиям. SaS метод основан на выборе внутри n-го слоя In отсчетных поверхностей параллельных срединной поверхности, чтобы ввести температуры, электрические потенциалы и перемещения этих поверхностей в качестве искомых функций. Такой выбор неизвестных с последующим использованием полиномов Лагранжа степени In -1 в аппроксимациях температуры, электрического потенциала, перемещений и механических параметров материала по толщине слоя приводит к эффективной модели термопьезоэлектрической оболочки. Внутренние SaS расположены внутри каждого слоя в узловых точках полинома Чебышева, что позволяет равномерно минимизировать погрешность приближения полиномами Лагранжа высокого порядка. В результате SaS метод может быть применен к построению аналитических решений для гибридных слоистых цилиндрических оболочек с пьезоэлектрическими сенсорами и актуаторами, которые асимптотически приближаются к точным решениям термопьезоэлектричества при стремлении числа отсчетных поверхностей In к бесконечности.
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Verwandte thermoelektromechanische Analyse von Hybrid-Zylinderschalen mit piezoelektrischen Sensoren und Aktuatoren
Zusammenfassung: Dieser Artikel befasst sich mit der Implementierung der Referenzoberflächenmethode (SaS-Methode) zur Lösung dreidimensional gekoppelter Probleme für geschichtete piezoelektrische Schalen, die Temperatur- und elektromechanischen Effekten ausgesetzt sind. Das SaS-Verfahren basiert auf der Auswahl von Bezugsflächen parallel zur mittleren Oberfläche innerhalb der n-ten Schicht In, um Temperaturen, elektrische Potentiale und Verschiebungen dieser
Oberflächen als gewünschte Funktionen einzuführen. Eine solche Auswahl von Unbekannten mit der anschließenden Verwendung von Lagrange-Polynomen des Grades In -1 in Approximationen von Temperatur, elektrischem Potential, Verschiebungen und mechanischen Parametern des Materials über die Schichtdicke führt zu einem effektiven Modell einer thermopiezoelektrischen Hülle. Interne SaSs befinden sich in jeder Schicht an den Knotenpunkten des Chebyshev-Polynoms, wodurch der Approximationsfehler von Lagrange-Polynomen hoher Ordnung gleichmäßig minimiert werden kann. Infolgedessen kann die SaS-Methode auf die Konstruktion von analytischen Lösungen für Hybridschicht-Zylinderschalen mit piezoelektrischen Sensoren und Aktuatoren angewendet werden, die sich asymptotisch den exakten Lösungen der Thermo-Piezoelektrizität nähern, wenn die Anzahl der Bezugsflächen In gegen unendlich tendiert.
Analyse associée thermoélectromécanique des coques cylindriques en couches hybrides avec capteurs et actionneurs piézoélectriques
Résumé: L'article est consacré à la mise en œuvre de la méthode SaS pour résoudre des problèmes liés 3D pour des couches piézoélectriques soumises à des influences thermiques et électromécaniques. La méthode SaS est basée sur la sélection à l'intérieur de la nième couche de surfaces de référence parallèles à la surface médiane pour introduire des températures, des potentiels électriques et le déplacement de ces surfaces en qualité des fonctions souhaitées. Un tel choix d'inconnues, suivie de l'utilisation des polynômes de Lagrange de degré dans les approximations de la température, du potentiel électrique, des déplacements et des paramètres mécaniques du matériau sur l'épaisseur de la couche, aboutit à un modèle efficace de la coque thermopyézoélectrique. Les SaS internes sont situés à l'intérieur de chaque couche aux points nodaux du polynôme de Chebyshev, ce qui permet de minimiser uniformément
l'erreur d'approximation des polynômes de Lagrange d'ordre élevé. En conséquence, la méthode SaS peut être appliquée à la construction de solutions analytiques pour des coques cylindriques en couches hybrides avec des capteurs piézoélectriques et des actuateurs qui se rapprochent asymptotiquement des solutions précises de thermopyézoélectricité tout en cherchant le nombre de surfaces de référence à l'infini.
Авторы: Куликов Геннадий Михайлович - доктор физико-математических наук, профессор, заведующий научно-исследовательской лабораторией «Механика интеллектуальных материалов и конструкций»; Меркушева Наталия Павловна -инженер научно-исследовательской лаборатории «Механика интеллектуальных материалов и конструкций»; Куликов Михаил Геннадьевич - кандидат технических наук, старший научный сотрудник научно-исследовательской лаборатории «Механика интеллектуальных материалов и конструкций»; Плотникова Светлана Валерьевна - доктор технических наук, ведущий научный сотрудник научно-исследовательской лаборатории «Механика интеллектуальных материалов и конструкций», ФГБОУ ВО «ТГТУ», г. Тамбов, Россия.
Рецензент: Нахман Александр Давидович - кандидат физико-математических наук, доцент кафедры «Высшая математика», ФГБОУ ВО «ТГТУ», г. Тамбов, Россия.
