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Effect of Convection, Internal Heat Source, and Solar Radiation on the Stress Analysis of a Rotating Functionally Graded Smart Disk

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Abstract

In this study, stresses and deformations of a rotating functionally graded magneto-electro-elastic (FGMEE) disk with non-uniform thickness considering internal heat generation, convection, and radiation heat transfer were investigated. The power-law function of the radial coordinate was considered for the properties of the material. Also, the heat conduction and convection coefficients are functions of temperature and radius. The heat transfer equation was derived considering thermal gradient, convection thermal boundary, heat source, and solar radiation. The differential transformation method (DTM) was used for solving the obtained nonlinear differential equation of heat transfer. Then, the equilibrium equation of the disk was derived and solved analytically. So, the radial stress, hoop stress, radial deformation, electric and magnetic potential, electric displacement, and magnetic induction can be obtained. Finally, some numerical examples were presented to examine the effects of the heat source, convection heat transfer, temperature dependency, solar radiation, inhomogeneity index, and angular velocity on the stress, deformation, electric displacement, and magnetic induction of the disk. The results show the tensile radial stress, deformation, electric displacement, and magnetic induction decrease for higher values of source power and solar flux intensity, while changes for the higher values of convection coefficient and thermal conductivity are opposite. Also, using a non-uniform thickness disk with an outer thickness smaller than the inner thickness can reduce the displacement and electromagnetic potentials.

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Abbreviations

\({r}_{o}\) :

Exterior radius \(\left(\mathrm{m}\right)\)

\({r}_{\mathrm{i}}\) :

Interior radius \(\left(\mathrm{m}\right)\)

\(T\) :

Temperature \(\left(\mathrm{K}\right)\)

\({T}_{\mathrm{a}}\) :

Ambient air temperature \(\left(\mathrm{K}\right)\)

\({T}_{b}\) :

Inner temperature \(\left(\mathrm{K}\right)\)

\({T}_{\mathrm{o}}\) :

Outer temperature \(\left(\mathrm{K}\right)\)

\({k}_{0}\) :

Thermal conductivity \(\left(\mathrm{W}/\mathrm{mK}\right)\)

\(\beta\) :

Conduction heat transfer coefficient

\(\omega\) :

Angular speed \(\left(\mathrm{Rad}/\mathrm{s}\right)\)

\({h}_{0}\) :

Convection heat transfer coefficient \(\left(\mathrm{W}/{\mathrm{m}}^{2}\mathrm{K}\right)\)

\({G}_{\mathrm{s}}\) :

Solar flux intensity \(\left(\mathrm{W}/{\mathrm{m}}^{2}\right)\)

\({\alpha }_{\mathrm{s}}\) :

Absorption coefficient of solar radiation

\(\varepsilon\) :

Emission coefficient of solar radiation

\(\psi\) :

Magnetic potential \(\left(\mathrm{A}\right)\)

\(\Omega\) :

Thickness profile coefficient

\({y}_{0}\) :

Thickness profile coefficient

\({a}_{i}\) :

Temperature function coefficients

\(\gamma\) :

Inhomogeneity index

\(a\) :

Convection coefficient

\(b\) :

Convection coefficient

\(\theta\) :

Dimensionless temperature

\(\eta\) :

Dimensionless radius

\({\sigma }_{\mathrm{r}}\) :

Radial stress \(\left(\mathrm{Pa}\right)\)

\({\sigma }_{\theta }\) :

Hoop stress \(\left(\mathrm{Pa}\right)\)

\({\sigma }_{z}\) :

Normal stress \(\left(\mathrm{Pa}\right)\)

\({u}_{\mathrm{r}}\) :

Radial displacement \(\left(\mathrm{m}\right)\)

\({c}_{ij}\) :

Elastic constants \(\left(\mathrm{GPa}\right)\)

\({D}_{r}\) :

Electric displacement \(\left(\mathrm{C}/{\mathrm{m}}^{2}\right)\)

\({e}_{ij}\) :

Piezoelectric coefficients \(\left(\mathrm{C}/{\mathrm{m}}^{2}\right)\)

\({q}_{ij}\) :

Piezomagnetic coefficients \(\left(\mathrm{N}/\mathrm{Am}\right)\)

\({\varepsilon }_{ij}\) :

Electromagnetic coefficients \(\left(\mathrm{Ns}/\mathrm{VC}\right)\)

\({\beta }_{ij}\) :

Dielectric coefficients \(\left({\mathrm{C}}^{2}/{\mathrm{Nm}}^{2}\right)\)

\({\alpha }_{i}\) :

Thermal expansion coefficients \(\left(1/\mathrm{K}\right)\)

\(\rho\) :

Density \(\left(\mathrm{kg}/{\mathrm{m}}^{3}\right)\)

\(\phi\) :

Electric potential \(\left(\mathrm{W}/\mathrm{A}\right)\)

\({B}_{r}\) :

Magnetic induction \(\left(\mathrm{T}\right)\)

\({\lambda }_{i}\) :

Thermal modulus \(\left(\mathrm{N}/{\mathrm{m}}^{2}\mathrm{K}\right)\)

\(\sigma\) :

Stefan–Boltzmann constant

\(H\{t\}\) :

DTM transformation coefficients

\({d}_{ij}\) :

Magnetic coefficients \(\left({\mathrm{Ns}}^{2}/{\mathrm{C}}^{2}\right)\)

\({\varepsilon }_{i}\) :

Components of strain

\({E}_{r}\) :

Components of electric field

\({p}_{i}\) :

Pyroelectric coefficients \(\left({\mathrm{C}}^{2}/{\mathrm{m}}^{2}\mathrm{K}\right)\)

\({m}_{i}\) :

Pyromagnetic coefficients \(\left(\mathrm{N}/\mathrm{AmK}\right)\)

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Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, University of Qom, Qom, Iran

    Mahdi Saadatfar, Mohammad Amin Babazadeh & Mojtaba Babaelahi

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  1. Mahdi Saadatfar
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  2. Mohammad Amin Babazadeh
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  3. Mojtaba Babaelahi
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Correspondence to Mahdi Saadatfar.

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Appendix

Appendix

The basic functions used in the DTM are defined as (Zhou 1986; Chen and Ho 1996):

See Table

Table 3 Changing functions in the DTM method
Full size table

3.

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Saadatfar, M., Babazadeh, M.A. & Babaelahi, M. Effect of Convection, Internal Heat Source, and Solar Radiation on the Stress Analysis of a Rotating Functionally Graded Smart Disk. Iran J Sci Technol Trans Mech Eng (2024). https://doi.org/10.1007/s40997-023-00725-y

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