Exact 3D Thermoelectroelastic Analysis of Piezoelectric Plates through a Sampling Surfaces Method

Journal of Intelligent Material Systems and Structures

2017

This article focuses on the implementation of the sampling surfaces method for the three-dimensional vibration analysis of layered piezoelectric plates. The sampling surfaces method is based on choosing inside the layers not equally spaced sampling surfaces parallel to the middle surface in order to introduce the displacements and electric potentials of these surfaces as basic plate variables. Such choice of unknowns with the consequent use of Lagrange polynomials in the assumed distributions of displacements, strains, electric potential, and electric field vector through the thicknesses of layers leads to the robust piezoelectric plate formulation. The sampling surfaces are located inside each layer at Chebyshev polynomial nodes that makes it possible to minimize uniformly the error due to the Lagrange interpolation. Therefore, the sampling surfaces formulation can be applied efficiently to the obtaining of benchmark solutions for the free vibration of layered piezoelectric plates, which asymptotically approach the three-dimensional solutions of piezoelectricity as the number of sampling surfaces tends to infinity.

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Section: Introductionmentioning

confidence: 99%

Benchmark solutions for the free vibration of layered piezoelectric plates based on a variational formulation

Journal of Intelligent Material Systems and Structures

2017

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“…It is seen from Equation (3) that the inner SaS

{normalΩ}^{true(n true) m_{n}}

are located at Chebyshev polynomial nodes (roots of the Chebyshev polynomial of degree I n − 2). This fact has a great meaning for the convergence of the SaS method 14,22 …”

Section: Higher‐order Heat Transfer Theory Of Laminated Shellsmentioning

confidence: 99%

“…Here, we introduce the second assumption of the SaS thermopiezoelectric shell formulation. Assume that the displacements

u_{i}^{(n)}

, strains

ϵ_{italicij}^{(n)}

, stresses

σ_{italicij}^{(n)}

, electric potential φ ( n ) , electric field

E_{i}^{(n)}

, electric displacements

D_{i}^{(n)}

, and entropy density S ( n ) are distributed through the thickness of the n th layer 14,22 as follows:

\begin{array}{l} [u_{i}^{true(n true)} ϵ_{ij}^{true(n true)} σ_{ij}^{true(n true)} φ^{true(n true)} E_{i}^{true(n true)} D_{i}^{true(n true)} S^{true(n true)}] \\ = \sum_{i_{n}} L^{(n) i_{n}} [u_{i}^{(n) i_{n}} ϵ_{ij}^{(n) i_{n}} σ_{ij}^{(n) i_{n}} φ^{(n) i_{n}} E_{i}^{(n) i_{n}} D_{i}^{(n) i_{n} <...>} \end{array}

…”

Section: Higher‐order Thermoelectroelastic Theory Of Laminated Shellsmentioning

confidence: 99%

“…Axisymmetric convective boundary conditions on the inner and outer surfaces 14 are written as

\begin{array}{l} Θ_{, 3}^{(n)} (- h false/ 2) - α^{-} Θ^{-} = 0, φ^{-} = σ_{33}^{-} = 0, \\ Θ_{, 3}^{(n)} (h false/ 2) + α^{+} Θ^{+} = α^{+} Θ_{0}, φ^{+} = σ_{33}^{+} = 0, \end{array}

where α − = h − / κ 33 , α + = h + / κ 33 , Θ 0 = 1 K, hα − = 0.2, hα + = 2, and n = 1.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…The 3D solution of thermoelectroelasticity for piezoelectric rectangular plates using the asymptotic expansion approach was obtained by Cheng and Batra 13 . The application of the SaS variational approach to the 3D‐coupled thermoelectroelastic analysis of laminated piezoelectric plates was carried out by the authors 14,15 . In the thermopiezoelectric shell formulation, the coefficients of the system of differential equations depend on the thickness coordinate.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Coupled thermoelectroelastic analysis of thick and thin laminated piezoelectric structures by exact geometry solid‐shell elements based on the sampling surfaces method

Numerical Meth Engineering

2021

A hybrid‐mixed exact geometry four‐node thermopiezoelectric solid‐shell element through the sampling surfaces (SaS) formulation is proposed. The SaS formulation is based on the choice of an arbitrary number of SaS within layers parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the temperatures, displacements and electric potentials of these surfaces as basic shell unknowns. Due to the variational formulation, the outer surfaces and interfaces are also included into a set of SaS. Such choice of unknowns with the use of Lagrange polynomials in the through‐thickness approximations of temperatures, temperature gradient, displacements, strains, electric potential, and electric field leads to a very compact higher‐order thermopiezoelectric shell formulation. To implement efficient analytical integration throughout the solid‐shell element, the extended assumed natural strain method is employed for all components of the temperature gradient, strain tensor, and electric field vector. The developed hybrid‐mixed four‐node thermopiezoelectric solid‐shell element is based on the Hu–Washizu variational principle and shows a superior performance in the case of coarse meshes. It can be useful for the three‐dimensional thermoelectroelastic analysis of thick and thin doubly curved laminated piezoelectric shells under thermal loading, since the SaS formulation allows one to obtain the numerical solutions with a prescribed accuracy, which asymptotically approach exact solutions of the theory of thermopiezoelectricity as a number of SaS tends to infinity.

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Coupled thermoelectroelastic stress analysis of piezoelectric shells

Mamontov

2015

Composite Structures