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Elastoplastic buckling of FGM beams in thermal environment

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Abstract

The elastoplastic buckling behaviors of functionally graded material (FGM) beams under axial compression loading are studied in consideration of temperature dependence of material properties. Firstly, elastoplastic material properties are obtained on the basis of bilinear hardening model and temperature dependence, meanwhile the elastoplastic constitutive equations of the FGM are established as well. Then introducing the Hamilton principle, the elastoplastic buckling behaviors of FGM beams are transformed into solving the eigenvalues in symplectic space. At the same time, the buckling critical loads corresponding to the generalized eigenvalues of the canonical equations can be calculated by the bifurcation conditions. Finally, the elastoplastic buckling characteristics and solution process of them are revealed by the symplectic method and effects of material gradient, geometrical parameters of structure and ambient temperature on the critical loads of elastoplastic buckling are discussed in detail.

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Abbreviations

A :

Cross section (m\(^{{2}})\)

\(A^{e}\) :

Cross-sectional area of elastic deformation zone (m\(^{{2}})\)

\(A^{p}\) :

Cross-sectional area of plastic flow zone (m\(^{{2}})\)

b :

Width (m)

B :

Stiffness coefficients

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

Constants

D :

Stiffness coefficients

E :

Young’s modulus (GPa)

g :

Stress–strain transfer ratio

\(g'\) :

Stress transfer coefficient

h :

Height (m)

H :

Tangent modulus (GPa)

\(\kappa \) :

Curvature

L :

Lagrange function

n :

Power law index

N :

Axial compression force (N)

\(P_{0} ,P_{-1} ,P_{1} ,P_{2} ,P_{3} \) :

Temperature dependence coefficients

P :

Material properties

Q :

Dimensionless deflection

s :

Location of the elastoplastic interface (m)

T :

Temperature (K)

uvwq :

Displacement components (m)

V :

Volume fractions

xyz :

Coordinatesx

\(\theta \) :

Dimensionless eigenvalue

\(\sigma _{x} \) :

Axial normal stress (MPa)

\(\sigma _{Y} \) :

Yield limit (MPa)

\(\sigma ^{e}\) :

Elastic stress

\(\sigma ^{p}\) :

Plastic stress

\(\varepsilon _{x},\varepsilon _{x}^{0} \) :

Strains

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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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Correspondence to Jinghua Zhang.

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Communicated by Andreas Öchsner.

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Zhang, J., Zheng, W. Elastoplastic buckling of FGM beams in thermal environment. Continuum Mech. Thermodyn. 33, 151–161 (2021). https://doi.org/10.1007/s00161-020-00895-z

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  • DOI: https://doi.org/10.1007/s00161-020-00895-z

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