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The photothermal interaction of a semiconducting solid sphere based on three Green-Naghdi theories due to the fractional-order strain and ramp-type heating

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Abstract

The present article deals with a new mathematical model of a thermoelastic semiconducting solid sphere based on the Green-Naghdi theories (type-I, type-II, and type-III) to study the photothermal interaction. The fractional-order strain consideration has been applied. The bounding surface of the sphere has been thermally loaded by ramp-type heating. Laplace’s transform has been carried out, and its inversions have been computed numerically using the Tzou method. The numerical results have been presented in figures with various values of the fractional order and ramp-time heat parameters. The fractional-order and ramp-time heat parameters significantly affect all the studied functions. The ramp-time heat parameter could be used to tune the thermomechanical energy among the interaction of the thermoelastic body due to thermal loading.

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Data Availability

All data generated or analyzed during this study are included in this published article.

Abbreviations

\(C_{E}\) :

Specific heat at constant strain

\(c_{o}\) :

\(= \sqrt{\frac{\lambda + 2 \mu }{\rho }} \) Longitudinal wave speed

\(D\) :

The mechanical damage variable parameter

\(D_{e}\) :

The coefficient of carrier diffusion

\(E_{g}\) :

The energy gap of the semiconductor

\(e_{ij}\) :

The strain components

\(K\) :

Thermal conductivity

\(K^{*}\) :

The characteristic constant of the Green-Naghdi model

\(k_{0}\) :

\(= \frac{\partial n_{0}}{\partial T}\) is the coupling parameter of thermal activation

\(F_{i}\) :

The body force components

\(S_{r}\) :

The speed of recombination on the surface

\(T\) :

The absolute temperature

\(T_{o}\) :

Reference temperature

\(t\) :

Time

\(u_{ij}\) :

The displacement components

\(\alpha _{n}\) :

Coefficient of linear electronic deformation

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

Coefficient of linear thermal expansion

\(\beta\) :

\(= \left ( \frac{\lambda + 2\mu }{\mu } \right )^{1 / 2}\)

\(\gamma _{t}\) :

\(= \left ( 3\lambda + 2\mu \right )\alpha _{T}\)

\(\gamma _{n}\) :

\(= \left ( 3\lambda + 2\mu \right )\alpha _{n}\)

η :

\(= \frac{\rho C_{E}}{K_{1}}\) The thermal viscosity

\(\lambda , \mu \) :

Lamé’s constants

\(\theta\) :

\(= \left ( T - T_{0} \right )\) Temperature increment

\(\rho \) :

Density

\(\sigma _{ij}\) :

Components of the stress tensor

\(\tau \) :

The lifetime of a photogenerated carrier

\(\tau _{0}\) :

The mechanical relaxation time

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt

    Hamdy M. Youssef

  2. Mechanical Engineering Department, College of Engineering and Islamic Architecture, Umm Al-Qura University, Makkah, Saudi Arabia

    Hamdy M. Youssef

  3. Mathematics Department, Al-Lith University College, Umm Al-Qura University, Al-Lith, Saudi Arabia

    Eman A. N. Al-Lehaibi

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  1. Hamdy M. Youssef
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  2. Eman A. N. Al-Lehaibi
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Correspondence to Hamdy M. Youssef.

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Youssef, H.M., Al-Lehaibi, E.A.N. The photothermal interaction of a semiconducting solid sphere based on three Green-Naghdi theories due to the fractional-order strain and ramp-type heating. Mech Time-Depend Mater 27, 1237–1256 (2023). https://doi.org/10.1007/s11043-022-09543-2

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