Stat-Space Approach to Three-Dimensional Thermoelastic Half-Space Based on Fractional Order Heat Conduction and Variable Thermal Conductivity Under Moor–Gibson–Thompson Theorem
Abstract
1. Introduction
2. The Governing Equations
3. The Problem Formulation
4. Applying the Laplace and Double Fourier Transforms
5. Formulation of the State-Space Approach
6. Inversion of the Laplace and the Double Fourier Transforms
7. Numerical Results and Discussion
8. Conclusions
- It is noted that the value of the fractional order parameter and the variability of the thermal conductivity have significant effects on the mechanical and thermal waves.
- The distributions of the temperature increment, invariant stress, and invariant strain fall to zero at a smaller distance from the bounding surface in the order of strong thermal conductivity, followed by normal thermal conductivity, and finally weak thermal conductivity.
- The Moor–Gibson–Thompson model is a successful model for simulating three-dimensional thermoelastic materials with variable thermal conductivity and fractional order heat conduction.
- Classifying the thermal conductivity as weak, normal, and strong is important and close to the physical behaviour of the thermal conductivity of the thermoelastic materials.
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
Time | |
Components of the stress tensor | |
Components of the displacement vector | |
Components of the strain tensor | |
Thermal conductivity | |
The thermal conductivity rate | |
The thermal conductivity rate at room temperature | |
The parameter of the thermal conductivity change | |
Absolute temperature | |
Reference temperature | |
Temperature increment | |
Specific heat at constant strain | |
The coefficient of linear thermal expansion | |
The diffusivity | |
Relaxation time due to heat flux lag | |
The average time of the thermal conductivity rate | |
Longitudinal wave speed | |
Lame’s constants | |
Density | |
The Kronecker delta function | |
The thermal viscosity | |
The ratio of the speed of the longitudinal wave to the shear wave | |
The fractional order parameter of heat conduction | |
The dimensionless thermoelastic coupling constant | |
The Laplacian operator |
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Youssef, H.M. Stat-Space Approach to Three-Dimensional Thermoelastic Half-Space Based on Fractional Order Heat Conduction and Variable Thermal Conductivity Under Moor–Gibson–Thompson Theorem. Fractal Fract. 2025, 9, 145. https://doi.org/10.3390/fractalfract9030145
Youssef HM. Stat-Space Approach to Three-Dimensional Thermoelastic Half-Space Based on Fractional Order Heat Conduction and Variable Thermal Conductivity Under Moor–Gibson–Thompson Theorem. Fractal and Fractional. 2025; 9(3):145. https://doi.org/10.3390/fractalfract9030145
Chicago/Turabian StyleYoussef, Hamdy M. 2025. "Stat-Space Approach to Three-Dimensional Thermoelastic Half-Space Based on Fractional Order Heat Conduction and Variable Thermal Conductivity Under Moor–Gibson–Thompson Theorem" Fractal and Fractional 9, no. 3: 145. https://doi.org/10.3390/fractalfract9030145
APA StyleYoussef, H. M. (2025). Stat-Space Approach to Three-Dimensional Thermoelastic Half-Space Based on Fractional Order Heat Conduction and Variable Thermal Conductivity Under Moor–Gibson–Thompson Theorem. Fractal and Fractional, 9(3), 145. https://doi.org/10.3390/fractalfract9030145