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Nonlinear in-plane thermal buckling of rotationally restrained functionally graded carbon nanotube reinforced composite shallow arches under uniform radial loading

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Abstract

The nonlinear in-plane instability of functionally graded carbon nanotube reinforced composite (FG-CNTRC) shallow circular arches with rotational constraints subject to a uniform radial load in a thermal environment is investigated. Assuming arches with thickness-graded material properties, four different distribution patterns of carbon nanotubes (CNTs) are considered. The classical arch theory and Donnell’s shallow shell theory assumptions are used to evaluate the arch displacement field, and the analytical solutions of buckling equilibrium equations and buckling loads are obtained by using the principle of virtual work. The critical geometric parameters are introduced to determine the criteria for buckling mode switching. Parametric studies are carried out to demonstrate the effects of temperature variations, material parameters, geometric parameters, and elastic constraints on the stability of the arch. It is found that increasing the volume fraction of CNTs and distributing CNTs away from the neutral axis significantly enhance the bending stiffness of the arch. In addition, the pretension and initial displacement caused by the temperature field have significant effects on the buckling behavior.

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Abbreviations

q :

uniform radial load

Θ:

half of the included angle of the arch

S :

length of the arch

b :

width of the arch

h :

thickness of the arch

R :

radius of the arch

k :

stiffness of equal elastic rotational restraints

ΔT :

uniform temperature rise

θ :

the angular coordinate

z :

coordinate

w, ̄w :

axial and dimensionless axial displacements

v, ̄v :

radial and dimensionless radial displacements

̄v*,̄w*:

radial and axial buckling displacements

̄v b, ̄w b :

radial and axial infinitesimal displacements

̄v tb, ̄w tb :

upper bifurcation buckling radial and axial displacements

Δ̄v, Δ̄w :

additional radial and axial displacements

Δ̄v sym, Δ̄v a :

symmetric and asymmetric components of radial displacements

V cnt :

total volume fraction of carbon nanotubes (CNTs)

w cnt :

mass fraction of CNTs

ρ cnt, ρ m :

densities of CNTs and the isotropic matrix

E cnt11 , E cnt22 :

Young’s moduli of CNTs

E m :

Young’s modulus of matrix

G cnt12 , G m :

shear moduli of CNTs and matrix

η 1, η 2, η 3 :

CNT efficiency parameters

ν :

Poisson’s ratio

α cnt11 , α m :

thermal expansion coefficients of CNTs and matrix

ε θ :

normal strain

ε M :

membrane strain

ε rr :

radial strain

ε T :

thermal strain

σ θθ :

circumferential normal strain

N :

axial compression force

N T :

thermal axial compression force

N * :

axial buckling force

N b :

axial infinitesimal force

N bb :

axial force during bifurcation buckling

N s :

axial force during the lowest buckling

M :

bending moment

M T :

thermal bending moment

M * :

bending moment for buckling

M b :

infinitesimal bending moment

A 11, B 11, D 11 :

stiffness components

κ :

effective bending stiffness

μ :

axial compressive force parameter

μ b :

axial force parameter during bifurcation buckling

μ s :

axial force parameter during the lowest buckling

\(\overline P\) :

dimensionless load

\(\overline P _{\rm{b}}^{\rm{u}}\) :

upper bifurcation buckling load

B :

stiffness parameter

B0 = B/h :

dimensionless stiffness parameter

ζ :

dimensionless temperature parameter

α :

flexibility of the rotational restraint

λ :

geometric parameter of the arch

λ sn :

critical geometric parameter for buckling

λ sb1 :

critical geometric parameter for the limit point and the bifurcation buckling mode switching

λ sb2 :

critical geometric parameter for bifurcation buckling

K :

amplitude parameter

β = μΘ:

parameter for calculation

β b, β s :

parameters for calculation

ρ :

temperature parameter for calculation

ρ 0 = ρ/h :

parameter for calculation

r 11 :

parameter for calculation

K 1, K 2 :

coefficients

A i, B i, C i :

coefficients, i=1, 2, 3, 4, 5

D i :

coefficients, i=1, 2, 3

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Authors and Affiliations

  1. School of Automotive Engineering, Changzhou Institute of Technology, Changzhou, Jiangsu Province, 213032, China

    Cheng Li

  2. School of Rail Transportation, Soochow University, Suzhou, Jiangsu Province, 215131, China

    Cheng Li, Chengxiu Zhu & Shuang Li

  3. Department of Architecture and Civil Engineering, City University of Hong Kong, Tat Chee Ave, Kowloon, Hong Kong Special Administrative Region, China

    C. W. Lim

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  1. Cheng Li
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  2. Chengxiu Zhu
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  3. C. W. Lim
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Correspondence to Shuang Li.

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Citation: LI, C., ZHU, C. X., LIM, C. W., and LI, S. Nonlinear in-plane thermal buckling of rotationally restrained functionally graded carbon nanotube reinforced composite shallow arches under uniform radial loading. Applied Mathematics and Mechanics (English Edition), 43(12), 1821–1840 (2022) https://doi.org/10.1007/s10483-022-2917-7

Project supported by the National Natural Science Foundation of China (Nos. 11972240 and 51875374)

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Li, C., Zhu, C., Lim, C.W. et al. Nonlinear in-plane thermal buckling of rotationally restrained functionally graded carbon nanotube reinforced composite shallow arches under uniform radial loading. Appl. Math. Mech.-Engl. Ed. 43, 1821–1840 (2022). https://doi.org/10.1007/s10483-022-2917-7

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