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Creep analysis in a rotating variable thickness functionally graded disc with convection heat transfer and heat source

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Abstract

The time-dependent creep behavior of a rotating disc composed of functionally graded material (FGM) with varying thickness was analyzed. The convection heat transfer along with internal heat generation was considered in thermoelastic analysis. The material properties were assumed to change radially as a power-law function. Also, the heat convection and heat conduction coefficients were taken as functions of temperature and radius. The nonlinear heat transfer equation was solved using the differential transformation method (DTM). Then, the equilibrium equation considering creep strains was derived. The derived differential equation was solved analytically for zero time. Considering the creep strains, the stress and strain rates were determined using Norton’s law with Prandtl–Reuss equations for steady-state thermal boundary conditions. Finally, the time-dependent creep stress redistributions at any time were evaluated using an iterative method. The effects of the heat source, convection heat transfer, temperature dependency, inhomogeneity index, and angular velocity on the behavior of the disc were explored in numerical examples.

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Abbreviations

\(r_{o}\) :

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

\(r_{i}\) :

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

\(H_{z}\) :

Magnetic field

\(T\) :

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

\(T_{a}\) :

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

\(T_{b}\) :

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

\(T_{o}\) :

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

\(k_{0}\) :

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

\(h_{0}\) :

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

\(q_{0}\) :

Internal heat generation \(\left ( {\mathrm{W}} / {\mathrm{m}^{3}} \right )\)

\(y_{0}\) :

Thickness profile coefficient

\(\Omega \) :

Thickness profile coefficient

\(e_{0}\) :

Internal heat generation coefficient

\(\gamma \) :

Grading index

\(\beta \) :

Conduction heat transfer coefficient

\(\omega \) :

Rotational-angular speed \(\left ( \mathrm{rad}/\mathrm{s} \right )\)

\(a\) :

Convection heat transfer coefficient

\(b\) :

Convection heat transfer coefficient

\(\theta \) :

Dimensionless temperature

\(\eta \) :

Dimensionless radius

\(\sigma _{r}\) :

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

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

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

\(\sigma _{e}\) :

von Mises equivalent stress \(\left ( \mathrm{Pa} \right )\)

\(u_{r}\) :

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

\(c_{ij}\) :

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

\(\alpha _{i}\) :

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

\(\rho \) :

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

\(\lambda _{i}\) :

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

\(\mu \) :

Magnetic permeability \(\left ( {\mathrm{N}} / {\mathrm{A}^{2}} \right )\)

\(\overrightarrow{h}\) :

Induced magnetic field

\(\overrightarrow{H}\) :

Magnetic field vector

\(\overrightarrow{J}\) :

Electric current density vector

\(\varepsilon _{i}\) :

Components of strain

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

Components of creep strain

\(\dot{\varepsilon}_{c}^{i}\) :

Components of creep strain rate \(\left ( 1/ \mathrm{s} \right )\)

\(\dot{\varepsilon}_{e}^{i}\) :

Effective creep strain rate \(\left ( 1/ \mathrm{s} \right )\)

\(f_{r}\) :

Lorentz force

\(a_{i}\) :

Temperature function coefficients

\(H \{t\}\) :

DTM transformation coefficients

\(dt\) :

Time increment \(\left ( \mathrm{s} \right )\)

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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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Contributions

All authors contributed to the study conception. Derivation of equations and solution were performed by Mahdi Saadatfar, Mohammad Amin Babazadeh and Mojtaba Babaelahi. The first draft of the manuscript was written by Mahdi Saadatfar and Mohammad Amin Babazadeh and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Mahdi Saadatfar.

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Appendix

Appendix

The changing functions used in the DTM are defined in Table 2 (Zhou (1986), Chen and Ho (1996)):

Table 2 Changing functions in the DTM method
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Saadatfar, M., Babazadeh, M.A. & Babaelahi, M. Creep analysis in a rotating variable thickness functionally graded disc with convection heat transfer and heat source. Mech Time-Depend Mater 28, 19–41 (2024). https://doi.org/10.1007/s11043-023-09613-z

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