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Dynamic buckling analysis of functionally graded material cylindrical shells under thermal shock

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Abstract

This study focuses on dynamic buckling of functionally graded material (FGM) cylindrical shells under thermal shock. The transient non-uniform temperature fields in the FGM shells subjected to dynamic thermal loading are determined using an analytic method. Based on the Hamiltonian principle, the dynamic thermal buckling problem of the FGM cylindrical shells is transformed into the symplectic space for solving. At the same time, the buckling thermal loads and buckling modes corresponding to generalized eigenvalues and eigen solutions of canonical equations can be calculated via the bifurcation conditions. The dynamic thermal buckling characteristics of the FGM cylindrical shells as well as the solving processes are given by the symplectic method. A complete dynamic buckling modes space is presented for the FGM cylindrical shells. The effects of the material gradient, parameters of structural geometry and thermal loadings on the dynamic buckling temperature are discussed.

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Abbreviations

L :

Length (m)

R :

Radius (m)

h :

Thickness (m)

P :

Material properties

E :

Young‘s modulus (GPa)

u, v, w :

Displacement components (m)

\(x,\theta ,z\) :

Coordinates

\(\rho \) :

Mass density (kg/m\(^{\mathrm {3}}\))

\(\alpha \) :

Thermal expansion coefficient (1/K)

C :

Thermal capacity [J/(kg K)]

K :

Thermal conductivity (W/mK)

V :

Volume fractions

k :

Power law index

\(\mu \) :

Poisson‘s ratio

\(\varepsilon _{xx} ,\varepsilon _{x\theta } ,\varepsilon _{\theta \theta }\) :

Strains

\(\sigma _{xx,} \sigma _{\theta \theta }\) :

Normal stresses (MPa)

\(\sigma _{x\theta }\) :

Shear stress (MPa)

\(\kappa _{xx}, \kappa _{\theta \theta }\) :

Curvature

\(\kappa _{x\theta }\) :

Torsional curvature

T :

Temperature (K)

t :

Time (s)

\(M_\mathrm{T}\) :

Bending moment (N m)

\(\Delta T\) :

Temperature rise (K)

\(T_{0}\) :

Initial temperature (K)

\(N_{xx} ,N_{\theta \theta } ,N_{x\theta }\) :

Membrane forces

\(M_{xx} ,M_{x\theta } ,M_{\theta \theta }\) :

Membrane moments

U :

Strain energy

\({\bar{U}}\) :

Density of strain energy

\({\bar{L}}\) :

Lagrange density function

H :

Hamiltonian function

\(\varvec{\upvarphi }\) :

State vector

\(\lambda _{n}\) :

Eigenvalue

\(\varvec{\upvarphi }_{n}\) :

Eigenvector

\(c_{1} ,c_{2} ,c_{3} ,c_{4}\) :

Constants

\(\lambda _{j}\) :

Eigen solutions

\(h_\mathrm{r}\) :

Heat exchange coefficient

\(\Delta T_\mathrm{{cr}}\) :

Critical temperatures (K)

\(\Delta t\) :

Increment of time (s)

m :

Axial wave

n :

Circumferential wave

\(\lambda \) :

Radius–thickness ratio

a :

Parameter of the load

\(N_\mathrm{T}\) :

Thermal membrane force (N)

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Acknowledgements

This work was supported by the National Natural Science Foundation of China [Grant Numbers 11662008, 11862012] and the abroad exchange funding for young backbone teachers of Lanzhou University of Technology.

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Authors and Affiliations

  1. School of Science, Lanzhou University of Technology, Lanzhou, 730050, Gansu, China

    Jinghua Zhang & Wu Zheng

  2. Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian, 116024, Liaoning, China

    Shuai Chen

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  1. Jinghua Zhang
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  2. Shuai Chen
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  3. Wu Zheng
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Correspondence to Jinghua Zhang.

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Communicated by Dr. Andreas Öchsner.

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Zhang, J., Chen, S. & Zheng, W. Dynamic buckling analysis of functionally graded material cylindrical shells under thermal shock. Continuum Mech. Thermodyn. 32, 1095–1108 (2020). https://doi.org/10.1007/s00161-019-00812-z

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