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Thermo-electro-mechanical vibration and buckling analysis of a functionally graded piezoelectric porous cylindrical microshell

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Abstract

In this paper, the vibration and buckling behavior of a functionally graded piezoelectric porous cylindrical microshell under thermo-electro-mechanical loads are explored on the basis of modified couple stress theory and higher-order shear deformation theory. First, the model of a functionally graded piezoelectric porous cylindrical microshell composed of piezoelectric materials with gradient change in the thickness direction was described. Second, the governing equations of the microshell were derived by Hamilton’s principle and Maxwell equation. Third, the modal frequency and buckling equations of the microshell with simply supported ends were obtained on the basis of harmonic trigonometric functions. Finally, the effects of parameters on the modal frequency and buckling behavior were carried out by case studies. The results show that the modal frequency of the microshell can be adjusted by changing the porosity volume fraction, power index, applied voltage, axial load and geometric dimensions. It is also found that the vibration of the microshell is suppressed by positive voltage and axial compression but is strengthened by negative voltage and axial tension, and the material length scale parameter increases stiffness. In addition, the effects of applied voltage and axial load on the buckling behavior are larger than those of temperature. Results can be used to guide the design and application of piezoelectric devices.

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Abbreviations

L :

Length of microshell

R :

Mid-surface radius of microshell

h :

Thickness of microshell

x :

Axial coordinate

θ :

Circumferential coordinate

y :

Tangential coordinate

z :

Radial coordinate

T i, T o :

Temperatures of inner and outer surfaces

T io :

Temperature difference

P(z):

Material properties of microshell

P i, P o :

Material properties of inner and outer surfaces

N :

Power index

α p :

Porosity volume fraction

u x :

Axial displacement component

u y :

Tangential displacement component

u z :

Radial displacement component

u :

Axial displacement component of any point on the mid-surface

v :

Tangential displacement component of any point on mid-surface

w :

Radial displacement component of any point on mid-surface

φ x, φ y :

Rotation angles of mid-surface normal around x and y axes

[C]:

Elastic moduli matrix

[β]:

Thermal moduli matrix

[d]:

Piezoelectric moduli matrix

[ξ]:

Dielectric moduli matrix

{σ}:

Stress vector

{ε}:

Strain vector

{E}:

Electric field vector

{D}:

Electric displacement vector

{p}:

Pyroelectric constants vector

σ xx, σ yy :

Normal stresses

T xy :

In-plane shear stress

T xz, T yz :

Shear stresses along thickness direction

ε xx, ε yy :

Normal strains

γ xy :

In-plane shear strain

γ xz, γ yz :

Shear strains along thickness direction

Dx, Dy, Dz :

Electric displacement components

E x, E y, E z :

Electric field components

c ije :

Equivalent elastic moduli components

ξ iie :

Equivalent dielectric moduli components

d ije :

Equivalent piezoelectric moduli components

β iie :

Equivalent thermal moduli components

p ie :

Equivalent pyroelectric constants components

Φ:

Electric potential inside the microshell

φ :

Electric potential variation

V :

Applied voltage

{m}:

Vector of higher-order stress tensor

{X}:

Vector of symmetric rotation gradient tensor

l :

Material length scale parameter

m ij :

Components of higher-order stress tensor

X ij :

Components of symmetric rotation gradient tensor

U :

Total strain energy

U c :

Strain energy based on CCT

U mc :

Strain energy based on MCST

K :

Kinetic energy

W :

Work done by external loads

\(N_{XX}^{T}\) :

Thermal load

N a :

Static axial load

\(N_{XX}^{p}\) :

Electric load

t :

Time variable

l 0, l 1, l 2, l 3, l 4, l 6 :

Generalized inertia constants

N ij, Q i :

Classical forces

M ij, S ij, J i :

Classical moments

Y ij :

Nonclassical forces

Γ ij, P ij, T ij :

Nonclassical moments

m :

Axial wavenumber

n :

Circumferential wavenumber

ω :

Modal frequency of microshell

[G]:

Mass matrix

{Δ}:

Displacement amplitude vector

{X}:

External load vector

[Z]:

Total stiffness matrix

[Z 0]:

Stiffness matrix

[Z N]:

Coefficient matrix of axial load

[Z T]:

Coefficient matrix of temperature difference

[Z V]:

Coefficient matrix of applied voltage

N cr :

Static buckling load

V cr :

Buckling voltage

Ω:

Dimensionless frequency

N d :

Dimensionless axial load

N acr :

Defined buckling load

N dcr :

Dimensionless buckling load

h/R :

Dimensionless thickness

L/R :

Dimensionless length

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Acknowledgments

This research is supported by the National Natural Science Foundation of China (Grant No. 51965042).

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Authors and Affiliations

  1. School of Aeronautical Manufacturing Engineering, Nanchang Hangkong University, Nanchang, Jiangxi Province, 330063, China

    Zhipeng Lyu, Wenguang Liu, Chao Liu, Yuhang Zhang & Mengxiang Fang

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  1. Zhipeng Lyu
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  2. Wenguang Liu
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Correspondence to Wenguang Liu.

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Wenguang Liu received his Ph.D. degree from Nanjing University of Aeronautics and Astronautics, China. He is currently working as an Associate Professor in Nanchang Hangkong University, China.

Zhipeng Lyu is currently studying as a graduating student in Nanchang Hangkong University, China. His research interests include multi-field coupling dynamics of piezoelectric structures.

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Lyu, Z., Liu, W., Liu, C. et al. Thermo-electro-mechanical vibration and buckling analysis of a functionally graded piezoelectric porous cylindrical microshell. J Mech Sci Technol 35, 4655–4672 (2021). https://doi.org/10.1007/s12206-021-0933-1

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  • DOI: https://doi.org/10.1007/s12206-021-0933-1

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