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Mathematical modeling of functionally graded nanobeams via fractional heat Conduction model with non-singular kernels

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Abstract

Functionally gradient materials (FGM) in nanobeams are interesting issues in the theory of elasticity and thermoelasticity regarding thermal and mechanical stress. These advanced heat-resistant materials are used as structural components in contemporary technology. The thermoelastic interactions in functionally graded nanobeams (FGN) have been studied in this article. The basic equations that control the introduced model have been established based on the Euler–Bernoulli beam concept, Eringen’s theory, and the two phase-lag fractional heat conduction model. The heat equation has been modeled and fractionalized into a new formula that includes nonsingular and nonlocal differential operators. The physical properties of the nanobeam vary in graded according to its thickness. The FGN nanobeam is subject to a time-dependent and periodically varying heat flow. The differential equations are analyzed analytically in the Laplace transform field. The responses in the nanobeam are graphically depicted for various fractional-order values, the influence of the nonlocal parameter and the periodic frequency of the heat flux. The results show that the gap between classical and nonlocal theories widens with increasing nonlocal parameters and decreasing nanobeam length.

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Abbreviations

λ, μ :

Lam’e’s constants

α m :

Thermal expansion coefficient

γ m :

Coupling parameter

T 0 :

Environmental temperature

θ = T−T 0 :

Temperature increment

T :

Absolute temperature

C Em :

Specific heat

e :

Cubical dilatation

σ ij :

Nonlocal stress tensor

e ij :

Strain tensor

L :

Length

A = bh :

Cross-sectional area

τ ij :

Local stress tensor

η :

Nonlocal parameter

P m :

Metal properties

ω :

Delay time

ν m :

Poisson’s ratio

τ θ :

Phase lag of temperature gradient

K m :

Thermal conductivity

α :

Fractional-order parameter

q i :

Components of the heat flux vector

δ ij :

Kronecker’s delta function

u i :

Displacement components

F i :

Body force components

Q :

Heat source

τ 0 :

Relaxation time

h :

Thickness

ρ m :

Density

b :

Width

oxyz:

Cartesian coordinate

∇2 :

Laplacian operator

E m :

Young’s modulus

P c :

Ceramic properties

t :

The time

χ m :

Thermal diffusivity

τ q :

Phase lag of heat flux

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Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at Jouf University for funding this work through research grant No. (DSR-2021-03-0379). We would also like to extend our sincere thanks to the College of Science and Arts in Al-Qurayyat for its technical support.

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  1. Department of Mathematics, College of Science and Arts, Al-Qurayat, Jouf University, Sakaka, Saudi Arabia

    Ahmed E. Abouelregal

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Abouelregal, A.E. Mathematical modeling of functionally graded nanobeams via fractional heat Conduction model with non-singular kernels. Arch Appl Mech 93, 977–995 (2023). https://doi.org/10.1007/s00419-022-02309-9

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