Investigation on Electrothermoelastic Behavior of FGPM Cylindrical Shells

Hong-Liang Dai; Yi-Nan Qi; Wei-Feng Luo

doi:10.1515/ijnsns-2015-0037

Article

Investigation on Electrothermoelastic Behavior of FGPM Cylindrical Shells

Hong-Liang Dai , Yi-Nan Qi and Wei-Feng Luo

Published/Copyright: January 15, 2016

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From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 17 Issue 1

Abstract

This paper presents an analytical solution for electrothermoelastic behavior of FGPM (functionally graded piezoelectric material) cylindrical shell. The cylindrical shell is assumed to be made up of two piezoelectric materials with their volume fractures varying along the thickness according to a simple power law. Based on classical thin shell theory, an analytical solution for electrothermoelastic performance of the FGPM cylindrical shell is presented. To investigate the influence of the power lower exponent, thermal environment, mechanical loading and electric boundary conditions on the electrothermoelastic behavior of FGPM cylindrical shells, numerical examples are presented and discussed, and some meaningful and valuable results are discovered, which will be very helpful for the design and application of such smart sensory structures.

Keywords: electrothermoelastic behavior; FGPM; cylindrical shell; analytical solution

MSC® (2010).: 74B20

Funding statement: Funding: The authors wish to thank reviewers for their valuable comments and the research is supported by the National Natural Science Foundation of China (11372105), New Century Excellent Talents Program in University (NCET-13-0184), State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body (71475004).

Nomenclature

c: the FGPMs are composed of two piezoelectric materials c and m, the subscript “c” denotes ceramic
m: subscript “m” denotes metal piezoelectric material of the FGPMs
Pc: material property of ceramic piezoelectric material of the FGPMs
Pm: material property of metal piezoelectric material of the FGPMs
Vc: volume fracture of ceramic piezoelectric material of the FGPMs
Vm: volume fracture of metal piezoelectric material of the FGPMs
Z: thickness of FGPMs [m], cylindrical coordinate system (x,y,z) is set on the mid-plane (z=0) of the left end, and the coordinate axes x, y and z are in the longitudinal, circumferential and inward radial directions, respectively
P(z): material properties of the FGPMs, P(z)=PcVc+PmVm, Vc+Vm=1
ci(z)(i=1,2): elastic constants
αi(z)(i=1,2): thermal expansion coefficients [1/K]
ei(z)(i=1,2): piezoelectric constants
g(z)(i=1,2): dielectric constants
T(z): temperature change [K], T(z)=T1+(T2−T1)∑i=051ni+1(Km−KcKm)i(12+zh)ni+1∑i=051ni+1(Km−KcKm)i
εi0（i=x, y）: strain components on the middle surface
ki0（i=x, y）: curvature changes on the middle surface
σi(i=1,2): components of stresses [N/m²]
Dz: radial electric displacement [C/m²]
ϕ: electric potential [W/A]
αi(z): (i = 1, 2) thermal expansion coefficients [1/K]
λi(i=1,2): function of thermal expansion coefficients, the expressions are λ1(z)=c1(z)α1(z)+c2(z)α2(z)λ2(z)=c2(z)α1(z)+c1(z)α2(z)
u(x): displacement along x axe [m]
w(x): displacement along z axe [m]
Qx: transversely shear force [N]
Nx, Ny, Mx: the membrane forces of shells [N] Nx=∫−h/2h/2σxdz,Ny=∫−h/2h/2σydz,Mx=∫−h/2h/2σxzdz

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Appendix A

G1(z)=c1(z)−e12(z)g(z),G2(z)=c2(z)−e1(z)e2(z)g(z),G3(z)=−p(z)e1(z)g(z)+λ1(z)T(z),G4(z)=e1(z)(R−z)g(z),G5(z)=c1(z)−e22(z)g(z),G6(z)=−p(z)e2(z)g(z)+λ2(z)T(z),G7(z)=e2(z)(R−z)g(z).

A1=∫−h/2h/2G1(z)dz,A2=∫−h/2h/2G1(z)⋅zdz,A3=∫−h/2h/2G2(z)dz,A4=∫−h/2h/2G2(z)⋅zdz,A5=∫−h/2h/2G3(z)dz,A6=∫−h/2h/2G4(z)dz,A7=∫−h/2h/2G5(z)dz,A8=∫−h/2h/2G5(z)⋅zdz,A9=∫−h/2h/2G6(z)dz,A10=∫−h/2h/2G7(z)dz,A11=∫−h/2h/2G1(z)⋅z2dz,A12=∫−h/2h/2G2(z)⋅z2dz,A13=∫−h/2h/2G3(z)⋅zdz,A14=∫−h/2h/2G4(z)⋅zdz.

Appendix B

ζ1=∫−h2ze1[z]g[z]dz, ζ2=∫−h2ze2[z]⋅zg[z]dz, ζ3=∫−h2ze2[z]g[z]dz, ζ4=∫−h2ze1[z]⋅zg[z]dz,ζ5=∫−h2z1g[z]⋅(R−z)dz, ζ6=∫−h2zp[z]⋅T[z]g[z]dz.

Received: 2015-4-6

Accepted: 2015-12-29

Published Online: 2016-1-15

Published in Print: 2016-2-1

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Articles in the same Issue

https://doi.org/10.1515/ijnsns-2015-0037

Keywords for this article

electrothermoelastic behavior; FGPM; cylindrical shell; analytical solution