2. The fractional thermoelastic model with two-fractional order
parameters
We know that, the classical Fourier’s law [1] which is given by
\(\overrightarrow{q}\left(x,t\right)=-K\nabla\theta\left(x,t\right)\)(1)
relates the heat flux vector\(\overrightarrow{q}\left(\mathbf{x},\ t\right)\) and the varying
temperature \(\theta=T-T_{0}\), where \(T\) is the absolute
temperature, \(T_{0}\) refers to the reference temperature and \(K\)indicates the thermal conductivity.
A model derived by Tzou [6-8] involving two-phase-lags\(({\tau_{q\ },\tau}_{\theta})\) is expressed as
\(\left(1+\tau_{q}\frac{\partial}{\partial t}\right)\overrightarrow{q}=-K\left(1+\tau_{\theta}\frac{\partial}{\partial t}\right)\nabla\theta\)(2)
Due to the significant applications of fractional derivatives in
different fields, we exchange the time derivatives by fractional
operators, and the two-phase lags \(\tau_{q\ },\tau_{\theta}\) by\(\tau_{q}^{\alpha}\) and \(\tau_{\theta}^{\beta}\). Therefore, Eq. (2)
can be rewritten in a fractional form as
\(\left(1+\tau_{q}^{\alpha}\frac{\partial^{\alpha}}{\partial t^{\alpha}}\right)\overrightarrow{q}=-K\left(1+\tau_{\theta}^{\beta}\frac{\partial^{\beta}}{\partial t^{\beta}}\right)\nabla\theta\)(3)
where \(\tau_{q}\geq\tau_{\theta}>0\), \(0<\alpha\ ,\ \beta<1\),
and \(\frac{\partial^{\alpha}}{\partial t^{\alpha}}\) is the
Riemann-Liouville fractional derivatives [13,14] defined by
\(\frac{\partial^{\alpha}}{\partial t^{\alpha}}f\left(t\right)=\frac{1}{\Gamma\left(1-\alpha\right)}\frac{d}{\text{dt}}\int_{0}^{t}\left(t-\xi\right)^{-\alpha}f\left(\xi\right)d\xi\)(4)
where \(\Gamma\left(\alpha\right)\) is the Gamma function. Noting
that, if \(f\left(t\right)\) is continuous, then
\(\lim_{\alpha\rightarrow 1}\left(\frac{d^{\alpha}}{dt^{\alpha}}f\left(t\right)\right)=f^{\prime}\left(t\right)\)(5)
The energy balance equation with heat source \(Q\) is given by [24,
25]
\(\rho C_{e}\frac{\partial\theta}{\partial t}+\gamma T_{0}\frac{\partial}{\partial t}\left(\operatorname{div}\overrightarrow{u}\right)-\rho Q=-\operatorname{div}\overrightarrow{q\ }\)(6)
where \(\gamma=\left(3\lambda+2\mu\right)\alpha_{t}\) is the
stress temperature modulus, \(\alpha_{t}\) is the thermal expansion
coefficient, \(C_{e}\) means the specific heat at constant strain,\(\lambda\), \(\mu\) are Lamé’s constants,\(\ \overrightarrow{u}\)refers to the displacement vector and \(\rho\) is the density of the
medium. Therefore, from Eqs. (3) and (6), we get
\(\left(1+\tau_{q}^{\alpha}\frac{\partial^{\alpha}}{\partial t^{\alpha}}\right)\left[\rho C_{e}\frac{\partial\theta}{\partial t}+\gamma T_{0}\frac{\partial}{\partial t}\left(\operatorname{div}\overrightarrow{u}\right)-\text{ρQ}\right]=K\left(1+\tau_{\theta}^{\beta}\frac{\partial^{\beta}}{\partial t^{\beta}}\right)\nabla^{2}\theta.\)(7)
This equation defines a generalized fractional model with two-phase-lags
and two fractional parameters \(\alpha,\ \beta\). We denoted the
modified model by 2FDPL-model. This constructed model has widely used in
wave propagation, modeling, biology, chemistry and viscosity
[10-11].
Finally, the additional basic equations of motion for a homogeneous and
isotropic thermoelastic solid are [24, 25]
\(\par
\begin{matrix}2e_{\text{ij}}=u_{j,i}+u_{i,j\text{\ \ }}\\
\mu u_{i,\text{jj}}+\left(\lambda+\mu\right)u_{j,\text{ij}}-\gamma\theta_{,i}+F_{i}=\rho{\ddot{u}}_{i}\\
\end{matrix}\) (8)
where \(F_{i\ }\)is the component of the external body forces.