# A Practical Approach for Thermal Stress of Functionally Graded Annular Fin

## Abstract

A practical approach is implemented for thermal stresses in an axisymmetric thin annular fin, made of functionally graded material. All material properties of the annular fin are assumed to be graded along the fin radius as a power-law function. A linear differential equation is derived to be the governing equation. Analytical solution of such equations except for simple grading functions is difficult or maybe not possible to implement for each parameter, so the numerical approach becomes inevitable. The novelty of the present study is to introduce the effects of mechanical and thermal properties on the thermal stress distribution of functionally graded annular fin with the help of a complementary function method (CFM). The complementary functions method will be incorporated into the analysis to convert the problem to an initial value problem, which can be easily solved by, for instance, Runge−Kutta methods with great accuracy. The results are validated for isotropic and homogeneous annular fin.

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## Abbreviations

a, b :

inner and outer radius of the fin

E 0 :

Young modulus of material of the fin at r = a

E(r):

Young modulus of material at any point of the fin

h :

heat transfer coefficient

k 0 :

thermal conductivity of material of the fin, r = a

k(r):

thermal conductivity of material at any point of the fin

N :

dimensionless parameter, N2 = 2ha2/(δ.k)0)

r :

R :

dimensionless outer radius, R = b/a

S r, S ϕ :

dimensionless radial and tangential stress, Sr = σr/E0, Sϕ = σϕ/E0

T :

temperature of the fin

T b :

base temperature of the fin

T :

ambient temperature

u :

ū :

dimensionless radial displacement, ū = u/a

α 0 :

linear thermal expansion coefficient of material of the fin at r = a

α(r):

linear thermal expansion coefficient of material at any point of the fin

β :

inhomogeneity parameters of Young modulus

γ :

inhomogeneity parameters of thermal conductivity

δ :

thickness of the fin

ε r, ε ϕ :

radial and tangential strain

θ :

dimensionless temperature, θ = (T - T)/(Tb - T)

λ :

inhomogeneity parameters of linear thermal expansion coefficient

ν :

Poisson’s ratio of material of the fin

ξ :

dimensionless radius, ξ = r/a

σ r, σ ϕ :

radial and tangential stress

ϕ :

tangential coordinate

χ:

dimensionless linear thermal expansion coefficient, χ = α0 (Tb) - T)

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Correspondence to K. Celebi.

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Yıldırım, A., Celebi, K. & Yarımpabuç, D. A Practical Approach for Thermal Stress of Functionally Graded Annular Fin. J. Engin. Thermophys. 28, 556–568 (2019). https://doi.org/10.1134/S1810232819040118