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Large deformation analysis of a cantilever beam made of axially functionally graded material by homotopy analysis method

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Abstract

Large deformation of a cantilever axially functionally graded (AFG) beam subject to a tip load is analytically studied using the homotopy analysis method (HAM). It is assumed that its Young’s modulus varies along the longitudinal direction according to a power law. Taking the solution of the corresponding homogeneous beam as the initial guess and obtaining a convergence region by adjusting an auxiliary parameter, the analytical expressions for large deformation of the AFG beam are provided. Results obtained by the HAM are compared with those obtained by the finite element method and those in the previous works to verify its validity. Good agreement is observed. A detailed parametric study is carried out. The results show that the axial material variation can greatly change the deformed configuration, which provides an approach to control and manage the deformation of beams. By tailoring the axial material distribution, a desired deformed configuration can be obtained for a specific load. The analytical solution presented herein can be a helpful tool for this procedure.

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Abbreviations

P :

tip load

L:

undeformed length

s :

arc length

E :

Young’s modulus

E R, E L :

values of E at right and left ends, respectively

I :

area moment of inertia

θ :

slope

q :

embedding parameter

N :

nonlinear operator

ψ :

solution to zeroth-order deformation equation

θ R, θ L :

solutions of beam with Young’s moduli Er and EL, respectively

a :

abscissa of end point

κ:

material gradient

α :

load parameter

β :

Young’s modulus ratio

θ 0 :

initial guess of slope

ξ:

ratio of arc length to beam length

L :

auxiliary operator

ℏ:

auxiliary parameter

H :

auxiliary function

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Authors and Affiliations

  1. School of Astronautics, Harbin Institute of Technology, Harbin, 150001, China

    Xin Lin & Yang Zhao

  2. School of Aerospace Engineering, Tsinghua University, Beijing, 100084, China

    Yixin Huang & Tianshu Wang

Authors
  1. Xin Lin
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  2. Yixin Huang
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  3. Yang Zhao
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  4. Tianshu Wang
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Corresponding author

Correspondence to Yixin Huang.

Additional information

Project supported by the China Postdoctoral Science Foundation (No. 2018M630167)

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Lin, X., Huang, Y., Zhao, Y. et al. Large deformation analysis of a cantilever beam made of axially functionally graded material by homotopy analysis method. Appl. Math. Mech.-Engl. Ed. 40, 1375–1386 (2019). https://doi.org/10.1007/s10483-019-2515-9

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  • DOI: https://doi.org/10.1007/s10483-019-2515-9

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

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