# Stresses distributions in a rotating functionally graded piezoelectric hollow cylinder

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## Abstract

An analytic solution to the axisymmetric problem of a long, radially polarized, hollow cylinder composed of functionally graded piezoelectric material (FGPM) rotating about its axis at a constant angular velocity is presented. For the case that electric, thermal and mechanical properties of the material obey different power laws in the thickness direction, distributions for radial displacement, stresses and electric potential in the FGPM hollow cylinder are determined by using the theory of electrothermoelasticity. Some useful discussions and numerical examples are presented to show the significant influence of material nonhomogeneity, and adopting suitable graded indexes and applying suitable geometric size and rotating velocity *ω* may optimize the rotating FGPM hollow cylindrical structures. This will be of particular importance in modern engineering application.

## Keywords

Stress distribution FGPM hollow cylinder Rotating Electrothermoelastic## Nomenclature

*u*radial displacement [m]

*r*radial variable [m]

*a*,*b*inner and outer radii of the FGPM hollow cylinder [m]

*σ*_{i}(*i*=*r*,*θ*,*z*)components of stresses [N/m

^{2}]*T*(*r*)temperature distribution [K]

*φ*(*r*)electric potential [W/A]

*D*_{r}radial electric displacement [C/m

^{2}]*c*_{ij}(*i*=1,2;*j*=1,2,3)elastic constant [N/m

^{2}]*e*_{1i}(*i*=1,2,3)piezoelectric constants [C/m

^{2}]*g*_{11}dielectric constant [C

^{2}/N m^{2}]*p*_{11}pyroelectric coefficient [C/m

^{2}K]*α*_{i}(*i*=1,2,3)thermal stress modulus [Pa/K]

*ρ*mass density [kg/m

^{3}]*ω*constant angular velocity of rotation [rad/s]

*k*thermal conduction coefficient [W/m K]

*h*_{a},*h*_{b}ratio of the convective heat-transfer coefficients [W/K]

## Non-dimensional quantities

- \(R = \frac{r - a}{b - a}\),
- \(u^{*} = \frac{u}{a}\),
- \(T^{*} = \frac{T(r)}{T_{0}}\),
- \(\sigma_{i}^{*} = \frac{\sigma _{i}}{P_{a}}\quad (i =r,\theta,z)\),
- \(\phi^{*} = \frac{\phi}{\phi _{a}}\)

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