Two-temperature generalized thermoelasticity with fractional order strain of an infinite body with a spherical cavity

Abstract

In this work, we constructed a mathematical model of an infinite elastic body with a spherical cavity in the context of the theory of generalized thermoelasticity with fractional order strain. The bounding plane of the cavity is connected with thermal shock. Laplace transform techniques are used, and the inversions are computed numerically using a method based on Fourier expansion techniques of Tzuo method. Some especial comparisons have been illustrated in figures to discuss the effects of the two-temperature and the fractional order parameters.

Keywords

Generalized thermoelasticity two-temperature fractional order Laplace transform

1. Introduction

The cartilage is a being part of the human quantity where it is a connective tissue that furnishes the rework for most of the junction in the body open a multi-ladder building that harnesses an extended order of single collagen and molecules to families of twisted macromolecular fibers and fibrils. To disentangle several-scatter modeling drive that prizes the macro-scale mechanical a quittance of cartilage from the micro-spread standard is the censure of the bioengineer so that this novel model will help them [1]. Fractional order Voigt standard performed better obtain to the integer direction models so, the unfolding recite here will help in better knowing the viscoelastic properties of soft human tissue and may direct to amended diagnostic applications [1]. Magin and Royston invent the first fork in which fractional calculus with the model of the insignificant usage sprain that characterizes the momentous action has been attaching. In this pattern, the compound with zero order is for a Hookean true and to one for a Newtonian fluid while the stretchable and the viscoelastic materials busy the order with a fractional order parameter between zero and one [1].

Youssef built the hypothesis, speculation of two-temperature generalized thermoelasticity with the general uniqueness theorem [2]. Youssef also derived a new theory of two-temperature generalized thermoelasticity theory for an isotropic and homogeneous thermoelastic body without energy dissipation [3] and induced the variational principle of this model [4].

Youssef derived a novel hypothesis of thermoelasticity supported on fraction order of sprain which is estimated as a new modification to Duhamel-Neumann of stress-strain description. After setting the equations which direct this speculation, Youssef explains the first applications of thermoelasticity with fractional order strain for one-dimensional and thermoelastic half-space based on different models of thermoelasticity [5]. Bassiouny and Youssef studied the thermoelastic behavior of a sandwich structure subjected to a thermal shock in context of generalized thermoelasticity with fractional order strain in the presence of a moving heat source [6]. Abbas has solved many problems in the context of two-temperature thermoelasticity theory [7, 8, 9, 10, 11].

In this paper, we will solve a mathematical model of an infinite thermoelastic body with a spherical cavity based on the theory of generalized thermoelasticity with fractional order strain. The bounding surface of the cavity is affected by a thermal shock. Laplace transform techniques will be applied, and the inversions will be computed numerically using a method based on Fourier expansion techniques of Tzuo iteration method.

2. The Governing equations

The equation of motion without body force takes the form [12]:

$\displaystyle\sigma_{ij,j}=\rho\ddot{u}_{i},\quad i,j=1,2,3.$ (1)

The constitutive equation takes the form [5]:

$\displaystyle\sigma_{ij}=\left(1+\tau^{\xi}D_{t}^{\xi}\right)\left(2\mu e_{ij}% +\lambda e_{kk}\delta_{ij}\right)-\gamma\left(T-T_{0}\right)\delta_{ij},\quad i% ,j=1,2,3.$ (2)

The heat conduction equation takes the form [5]:

$\displaystyle K\varphi_{,ii}=\rho C_{E}\left(\frac{\partial}{\partial t}+\tau_% {0}\frac{\partial^{2}}{\partial t^{2}}\right)T+\gamma T_{0}\left(\frac{% \partial}{\partial t}+\tau_{0}\frac{\partial^{2}}{\partial t^{2}}\right)\left(% 1+\tau^{\xi}D_{t}^{\xi}\right)e,\quad i=1,2,3.$ (3)

Where the fractional derivative with respect to the time $D_{t}^{\beta}$ is defined as [5]:

$\displaystyle D_{t}^{\xi}f(t)=\left\{\begin{array}[]{ll}\frac{1}{\Gamma(1-\xi)% }\int\limits_{0}^{t}\frac{1}{(t-\tau)^{\xi}}\frac{df(\tau)}{d\tau}d\tau&\quad 0% \leqslant\xi<1\\ \frac{df(\tau)}{d\tau}&\quad\xi=1\end{array}\right\}.$ (4)

The relation between the heat conduction and the dynamical heat takes the form [13]:

$\displaystyle\varphi-T=a\varphi_{,ii},\quad i=1,2,3.$ (5)

where $a$ is a non-negative parameter is called two-temperature parameter.

We will suppose elastic and homogenous infinite body with a spherical cavity with radius $R$ , i.e., the medium occupies the region $R\leqslant r<\infty$ and obeys the Eqs (1)–(5) and initially quiescent. Because of the symmetry, all the state functions are depending only on the dimension $r$ and the time $t$ . Thus, the displacement components for one dimension medium in spherical coordinates $(r,\phi,\psi)$ have the form [14]:

$\displaystyle u_{r}=u(r,t),\quad u_{\phi}=u_{\psi}=0$ (6)

The strain component takes the form

$\displaystyle e_{rr}=\frac{\partial u}{\partial r},\quad e_{\phi\phi}=e_{\psi% \psi}=\frac{u}{r},\quad e_{r\phi}=e_{r\psi}=e_{\phi\psi}=0,$ (7)

and

$\displaystyle e=e_{rr}+e_{\phi\phi}+e_{\psi\psi},$ (8)

Hence, we have

$\displaystyle e=\frac{\partial u}{\partial r}+2\frac{u}{r}=\frac{1}{r^{2}}% \frac{\partial(r^{2}u)}{\partial r}$ (9)

The equation of motion takes the form

$\displaystyle\frac{\partial\sigma_{rr}}{\partial r}=\rho\ddot{u},$ (10)

The constitutive equations take the forms

$\displaystyle\sigma_{rr}=2\mu\left(1+\tau^{\xi}D_{t}^{\xi}\right)\frac{% \partial u}{\partial r}+\lambda\left(1+\tau^{\xi}D_{t}^{\xi}\right)e-\gamma(T-% T_{0}),$ (11)

and

$\displaystyle\sigma_{\phi\phi}=\sigma_{\psi\psi}=2\mu\left(1+\tau^{\xi}D_{t}^{% \xi}\right)\frac{u}{r}+\lambda\left(1+\tau^{\xi}D_{t}^{\xi}\right)e-\gamma(T-T% _{0}).$ (12)

The heat conduction equation takes the form ,

$\displaystyle K\nabla^{2}\varphi=\rho C_{E}\left(\frac{\partial}{\partial t}+% \tau_{0}\frac{\partial^{2}}{\partial t^{2}}\right)T+\gamma T_{0}\left(\frac{% \partial}{\partial t}+\tau_{0}\frac{\partial^{2}}{\partial t^{2}}\right)\left(% 1+\tau^{\xi}D_{t}^{\xi}\right)e,$ (13)

The relation between the heat conduction and dynamical heat takes the form

$\displaystyle\varphi-T=a\nabla^{2}\varphi.$ (14)

where $\nabla^{2}=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial% }{\partial r}\right)$ .

Equation (10) could be written as follows:

$\displaystyle\rho\ddot{u}=2\mu\left(1+\tau^{\xi}D_{t}^{\xi}\right)\frac{% \partial^{2}u}{\partial r^{2}}+\lambda\left(1+\tau^{\xi}D_{t}^{\xi}\right)% \frac{\partial e}{\partial r}-\gamma\frac{\partial T}{\partial r}.$ (15)

For simplicity, we will use the following non-dimensional variables [15, 16]:

$\displaystyle r^{\prime}=c_{0}\eta r,\;u^{\prime}=c_{0}\eta u,\;\tau^{\prime}_% {0}=c_{0}^{2}\eta\tau_{0},\;t_{0}=c_{0}^{2}\eta t,\;\theta=\frac{T-T_{0}}{T_{0% }},\;\varphi^{\prime}=\frac{\varphi-T_{0}}{T_{0}},\;\sigma^{\prime}=\frac{% \sigma}{2\mu+\lambda},$ (16)

where $c_{0}^{2}=\left(\frac{2\mu+\lambda}{\rho}\right)$ and $\eta=\frac{\rho C_{E}}{K}$ .

Hence, after dropping the primes, we have

$\displaystyle\ddot{u}=\left(1+\tau^{\xi}D_{t}^{\xi}\right)\frac{\partial e}{% \partial r}-\alpha\frac{\partial\theta}{\partial r},$ (17)

Multiply Eq. (18) by $r^{2}$ and differentiate with respect to $r$ then divided by $r^{2}$ , we obtain

$\displaystyle\ddot{e}=\left(1+\tau^{\xi}D_{t}^{\xi}\right)\nabla^{2}e-\alpha% \nabla^{2}\theta.$ (18)

The heat equations take the forms

$\displaystyle\nabla^{2}\varphi=\left(\frac{\partial}{\partial t}+\tau_{0}\frac% {\partial^{2}}{\partial t^{2}}\right)\theta+\varepsilon\left(\frac{\partial}{% \partial t}+\tau_{0}\frac{\partial^{2}}{\partial t^{2}}\right)\left(1+\tau^{% \xi}D_{t}^{\xi}\right)e,$ (19)

and

$\displaystyle\varphi-\theta=\omega\nabla^{2}\varphi.$ (20)

Equations (11) and (12) take the forms:

$\displaystyle\sigma_{rr}=\beta^{2}\left(1+\tau^{\xi}D_{t}^{\xi}\right)e+(1-% \beta^{2})\left(1+\tau^{\xi}D_{t}^{\xi}\right)\frac{\partial u}{\partial r}-% \alpha\theta,$ (21)

and

$\displaystyle\sigma_{\phi\phi}=\sigma_{\psi\psi}=(1-\beta^{2})\left(1+\tau^{% \xi}D_{t}^{\xi}\right)\frac{u}{r}+\beta^{2}\left(1+\tau^{\xi}D_{t}^{\xi}\right% )e-\alpha\theta,$ (22)

where $\varepsilon=\frac{\gamma}{\rho C_{E}}$ , $\alpha=\frac{\gamma T_{0}}{\lambda+2\mu}$ and $\omega=ac_{0}^{2}\eta^{2}$ .

Applying the Laplace transform for both sides of the Eqs (18)–(22), this is defined as follows:

$\displaystyle\bar{f}(s)=\int\limits_{0}^{\infty}f(t)e^{-st}dt.$

Hence, we obtain

$\displaystyle s^{2}\bar{e}=\left(1+\tau^{\xi}s^{\xi}\right)\nabla^{2}\bar{e}-% \alpha\nabla^{2}\bar{\theta},$ (23) $\displaystyle\nabla^{2}\bar{\varphi}=(s+\tau_{0}s^{2})\bar{\theta}+\varepsilon% (s+\tau_{0}s^{2})\left(1+\tau^{\xi}s^{\xi}\right)\bar{e},$ (24) $\displaystyle\bar{\varphi}-\bar{\theta}=\omega\nabla^{2}\bar{\varphi},$ (25) $\displaystyle\bar{\sigma}_{rr}=\beta^{2}\left(1+\tau^{\xi}s^{\xi}\right)\bar{e% }+(1-\beta^{2})\left(1+\tau^{\xi}s^{\xi}\right)\frac{\partial\bar{u}}{\partial r% }-\alpha\bar{\theta},$ (26) $\displaystyle\bar{\sigma}_{\phi\phi}=\bar{\sigma}_{\psi\psi}=(1-\beta^{2})% \left(1+\tau^{\xi}s^{\xi}\right)\frac{\bar{u}}{r}+\beta^{2}\left(1+\tau^{\xi}s% ^{\xi}\right)\bar{e}-\alpha\bar{\theta},$ (27) $\displaystyle\bar{e}=\frac{\partial\bar{u}}{\partial r}+2\frac{\bar{u}}{r}=% \frac{1}{r^{2}}\frac{\partial(r^{2}\bar{u})}{\partial r},$ (28)

where the rule for the Laplace transforms of the Riemann-Liouville fractional derivative, reads from [17]:

$\displaystyle L\left\{D_{t}^{\xi}f(t)\right\}=s^{\xi}L\left\{f(t)\right\}=s^{% \xi}\bar{f}(s)\quad\xi>0,$ (29)

where the initial conditions are:

$\displaystyle\left.u(r,t)\right|_{t=0}=\left.\varphi(r,t)\right|_{t=0}=\left.% \theta(r,t)\right|_{t=0}=0$ (30) $\displaystyle\left.\frac{\partial u(r,t)}{\partial t}\right|_{t=0}=\left.\frac% {\partial\varphi(r,t)}{\partial t}\right|_{t=0}=\left.\frac{\partial\theta(r,t% )}{\partial t}\right|_{t=0}=0.$

From Eqs (24) and (25), we get

$\displaystyle\bar{\theta}=\alpha_{1}\bar{\varphi}-\alpha_{2}\bar{e},$ (31)

where

$\displaystyle\alpha_{1}=\frac{1}{1+\omega\left(s+\tau_{0}s^{2}\right)},\alpha_% {2}=\frac{\omega\varepsilon\left(1+\tau^{\xi}s^{\xi}\right)\left(s+\tau_{0}s^{% 2}\right)}{1+\omega\left(s+\tau_{0}s^{2}\right)}.$

Substituting from Eq. (31) into Eqs (23) and (24), we obtain

$\displaystyle\nabla^{2}\bar{\varphi}=\alpha_{3}\bar{\varphi}+\alpha_{4}\bar{e},$ (32)

and

$\displaystyle\nabla^{2}\bar{e}=\alpha_{5}\bar{\varphi}+\alpha_{6}\bar{e},$ (33)

where

$\displaystyle\alpha_{3}=\frac{\left(s+\tau_{0}s^{2}\right)}{1+\omega\left(s+% \tau_{0}s^{2}\right)},\quad\alpha_{4}=\frac{\varepsilon\left(1+\tau^{\xi}s^{% \xi}\right)\left(s+\tau_{0}s^{2}\right)}{1+\omega\left(s+\tau_{0}s^{2}\right)},$ $\displaystyle\alpha_{5}=\frac{\alpha\alpha_{1}\alpha_{3}}{\left(1+\tau^{\xi}s^% {\xi}\right)+\alpha\alpha_{2}},\quad\alpha_{6}=\frac{s^{2}+\alpha\alpha_{1}% \alpha_{4}}{\left(1+\tau^{\xi}s^{\xi}\right)+\alpha\alpha_{2}},$

From Eqs (32) and (33), we have

$\displaystyle\left[\nabla^{4}-(\alpha_{3}+\alpha_{6})\nabla^{2}+(\alpha_{3}% \alpha_{6}-\alpha_{4}\alpha_{5})\right]\left\{\bar{\varphi}(r,s),\bar{e}(r,s)% \right\}=0.$ (34)

The general solution of Eq. (34) takes the forms

$\displaystyle\bar{\varphi}(r,s)=C_{1}\frac{e^{-k_{1}r}}{r}+C_{2}\frac{e^{-k_{2% }r}}{r},\quad R\leqslant r<\infty,$ (35)

and

$\displaystyle\bar{e}(r,s)=\frac{C_{1}(k_{1}^{2}-\alpha_{3})}{\alpha_{4}}\frac{% e^{-k_{1}r}}{r}+\frac{C_{2}(k_{2}^{2}-\alpha_{3})}{\alpha_{4}}\frac{e^{-k_{2}r% }}{r},\quad R\leqslant r<\infty,$ (36)

where $C_{1}=C_{1}(s),C_{2}=C_{2}(s)$ are parameters and $\pm k_{1},\pm k_{2}$ are the roots of the equation

$\displaystyle k^{4}-(\alpha_{3}+\alpha_{6})k^{2}+(\alpha_{3}\alpha_{6}-\alpha_% {4}\alpha_{5})=0.$ (37)

Hence, we have the following relations

$\displaystyle k_{1}^{2}+k_{2}^{2}=\alpha_{3}+\alpha_{6},\quad k_{1}^{2}k_{2}^{% 2}=\alpha_{3}\alpha_{6}-\alpha_{4}\alpha_{5}.$ (38)

In the Eqs (34) and (35) the positive powers of the expositions have been canceled to make the solutions bounded fora considerable value of $r$ , i.e. $\bar{\varphi}(r,s)|_{r\to\infty}=\bar{e}(r,s)|_{r\to\infty}=0$ .

To get the values of the parameters $C_{1}$ and $C_{2}$ we consider that the bounding plane of the cavity $r=R$ is subjected to a thermal loading in the following form

$\displaystyle\varphi(R,t)=\varphi_{R}=F(t).$ (39)

Applying Laplace transform, we have

$\displaystyle\bar{\varphi}(R,s)=\bar{\varphi}_{R}=\bar{F}(s).$ (40)

Moreover, we consider also that the bounding plane of the cavity $r=R$ has a mechanical loading which is enough to prevent the cubical dilatation, i.e.

$\displaystyle e(R,t)=e_{R}=0.$ (41)

Applying Laplace transform, we obtain

$\displaystyle\bar{e}(R,s)=\bar{e}_{R}=0.$ (42)

Hence, we have the following system:

$\displaystyle C_{1}e^{-k_{1}R}+C_{2}e^{-k_{2}R}=R\bar{\varphi}_{R},$ (43)

and

$\displaystyle C_{1}(k_{1}^{2}-\alpha_{3})e^{-k_{1}R}+C_{2}(k_{1}^{2}-\alpha_{3% })e^{-k_{2}R}=0,$ (44)

Figure 1.

The conductive temperature distribution.

Figure 2.

The dynamical temperature distribution.

Solving the system on Eqs (40) and (41) gives

$\displaystyle C_{1}=\frac{\bar{\varphi}_{R}\text{Re}^{k_{1}R}(k_{1}^{2}-\alpha% _{6})}{k_{1}^{2}-k_{2}^{2}},\quad C_{2}=\frac{\bar{\varphi}_{R}\text{Re}^{k_{2% }R}(k_{2}^{2}-\alpha_{6})}{k_{1}^{2}-k_{2}^{2}}.$ (45)

Finally, we have the solutions as follows:

$\displaystyle\bar{\varphi}(r,s)=\frac{\bar{\varphi}_{R}\text{R}}{(k_{1}^{2}-k_% {2}^{2})}\left[(k_{1}^{2}-\alpha_{6})\frac{e^{-k_{1}(r-R)}}{r}-(k_{2}^{2}-% \alpha_{6})\frac{e^{-k_{2}(r-R)}}{r}\right],$ (46) $\displaystyle\bar{e}(r,s)=\frac{\bar{\varphi}_{R}\text{R}\alpha_{5}}{(k_{1}^{2% }-k_{2}^{2})}\left[\frac{e^{-k_{1}(r-R)}}{r}-\frac{e^{-k_{2}(r-R)}}{r}\right],$ (47) $\displaystyle\bar{\theta}(r,s)=\frac{\bar{\varphi}_{R}\text{R}}{(k_{1}^{2}-k_{% 2}^{2})}\left[(k_{1}^{2}-\alpha_{6})(1-\omega k_{1}^{2})\frac{e^{-k_{1}(r-R)}}% {r}-(k_{2}^{2}-\alpha_{6})(1-\omega k_{2}^{2})\frac{e^{-k_{2}(r-R)}}{r}\right],$ (48) $\displaystyle\bar{u}(r,s)=\frac{\bar{\varphi}_{R}\text{R}\alpha_{5}}{(k_{1}^{2% }-k_{2}^{2})}\left[\frac{(1+k_{1}r)e^{-k_{1}(r-R)}}{k_{1}^{2}r^{2}}-\frac{(1+k% _{2}r)e^{-k_{2}(r-R)}}{k_{2}^{2}r^{2}}\right].$ (49)

We consider the thermal loading of the bounding surface of the cavity in the thermal shock form as follows:

$\displaystyle\varphi(R,t)=\varphi_{R}=F(t)=\varphi_{o}H(t),$ (50)

where $H(t)$ is called the Heaviside unit step function and $\varphi_{o}$ is constant which is the intensity of the thermal shock.

Hence, we have

$\displaystyle\bar{\varphi}_{R}=\frac{\varphi_{o}}{s},$ (51)

which complete the solution in the Laplace transform domain.

3. Inverse Laplace transforms

The solutions in the time domain and the numerical results can be calculated by using the Riemann-sum approximation method. Any function in Laplace domain can be inverted to the time domain as [18]:

$\displaystyle f(t)=\frac{e^{\kappa t}}{t}\left[\frac{1}{2}\bar{f}(\kappa)+% \text{Re}\sum\limits_{n=1}^{N}(-1)^{n}\bar{f}\left(\kappa+\frac{in\pi}{t}% \right)\right],$ (52)

where Re is the real part and is imaginary number unit. For faster convergence, many results of numerical experiments shown that the value of parameter $\kappa$ runs with the relation $\kappa t\approx$ 4.7 [18].

4. Numerical results and discussion

The copper significant was preferred for the aim of numerical evaluations, and the constants of the proposition were taken as follows [6, 12, 14, 15, 16, 19]:

K $=$ 386 N/K.sec, $\alpha_{T}=$ 1.78(10) ${}^{-5}$ K ${}^{-1}$ , $C_{E}=$ 383.1 m ${}^{2}$ /K, $\eta=$ 8886.73 m/sec ${}^{2}$ , $\rho=$ 8954 kg/m ${}^{3}$ , $\mu=$ 3.86(10) ${}^{10}$ N/m ${}^{2}$ , $\lambda=$ 7.76(10) ${}^{10}$ N/m ${}^{2}$ , $\tau_{0}=$ 0.13 $\times$ 10 ${}^{-14}$ sec, $\tau=$ 0.33 $\times$ 10 ${}^{-14}$ sec, $T_{0}=$ 293 K, $a=$ 0.73 $\times$ 10 ${}^{-18}$ m ${}^{-2}$ .

The computations were carried out for the non-dimensional time $t=$ 0.2, the radius of the cavity $R=$ 1.0, the intensity of the thermal shock $\varphi_{o}=$ 1.0, $\varepsilon=$ 1.60862, $\omega=$ 0.1, and $\alpha=$ 0.010444.

The conductive temperature, the dynamical temperature, the strain, the displacement, and the radial stress are represented graphically with various values of the non-dimensional distance $r$ ( $R\leqslant r\leqslant 2.0$ ).

Figures 1 and 2 represent the conductive and the dynamical temperature, respectively, for the one-temperature ( $\omega=$ 0) and the two-temperature ( $\omega=$ 0.1) models without fractional order strain ( $\tau=$ 0.0) and with fractional order strain ( $\tau=$ 0.05, $\xi=$ 0.5) to discuss the effect of the two-temperature and the fractional order parameters on the conductive and the dynamical temperature distribution. We found that the two-temperature parameter $\omega$ has a significant effect on the conductive and the dynamical temperature increment distribution, while the fractional order parameter does not affect.

Figures 3–5 represent the strain, the displacement, and the radial stress, respectively, for the one-temperature ( $\omega=$ 0) and the two-temperature ( $\omega=$ 0.1) models without fractional order strain ( $\tau=$ 0.0) and with fractional order strain ( $\tau=$ 0.05, $\xi=$ 0.5) to study the effects of the two-temperature and the fractional order parameters. We found that each of the two-temperature parameter and the fractional order parameter has a significant effect on the strain, the displacement, and the radial stress. The values of the peaks increase for the models of the two-temperature and the fractional order strain and decrease for the models of the one-temperature and without fractional order strain.

Figure 3.

The strain distribution.

Figure 4.

The displacement distribution.

Figure 5.

The radial stress distribution.

Figures 6 and 7 represent the conductive and the dynamical temperature, respectively, for the two-temperature model ( $\omega=$ 0.1) and fractional order strain ( $\tau=$ 0.05, 0.0 $\leqslant\xi\leqslant$ 1.0) to stand on the effect the fractional order parameter on the thermal waves. We found that the fractional order parameter does not affect the thermal waves.

Figure 6.

The conductive temperature with various fraction order parameter.

Figure 7.

The dynamical temperature with various fraction order parameter.

Figures 8–10 represent the strain, the displacement, and the radial stress, respectively, for the two-temperature model ( $\omega=$ 0.1) and fractional order strain ( $\tau=$ 0.05, 0.0 $\leqslant\xi\leqslant$ 1.0) to know the effect of the fractional order parameter. We found that the fractional order parameter has significant effects on the strain, the displacement, and the radial stress distributions.

Figure 8.

The strain with various fraction order parameter.

Figure 9.

The displacement with various fraction order parameter.

Figure 10.

The radial stress with various fraction order parameter.

5. Conclusion

A mathematical model of an infinite elastic body with a spherical cavity in the context of the theory of generalized thermoelasticity with fractional order strain has been constructed. The medium is assumed initially quiescent, and the bounding plane of the cavity is subjected to thermal shock. The results of this work show that the fractional order parameter and the two-temperature parameter have significant effects on the thermal and the mechanical waves.

Footnotes

Nomenclature

References

Magin

R.L.

and Royston

T.J.

, Fractional-order elastic models of cartilage: A multi-scale approach, Communications in Nonlinear Science and Numerical Simulation 15 (2010), 657–664.

Youssef

H.M.

, Theory of two-temperature-generalized thermoelasticity, IMA Journal of Applied Mathematics 71 (2006), 383–390.

Youssef

H.M.

, Theory of two-temperature thermoelasticity without energy dissipation, Journal of Thermal Stresses 34 (2011), 138–146.

Youssef

H.M.

, Variational principle of two-temperature thermoelasticity without energy dissipation, Journal of Thermoelasticity 1 (2013), 42–44.

Youssef

H.M.

, Theory of generalized thermoelasticity with fractional order strain, Journal of Vibration and Control 22 (2016), 3840–3857.

Bassiouny

and Youssef

H.M.

, Sandwich structure panel subjected to thermal loading using fractional order equation of motion and moving heat source, Canadian Journal of Physics 96 (2018), 174–182.

Abbas

I.A.

, A GN model based upon two-temperature generalized thermoelastic theory in an unbounded medium with a spherical cavity, Applied Mathematics and Computation 245 (2014), 108–115.

Abbas

I.A.

, Eigenvalue approach for an unbounded medium with a spherical cavity based upon two-temperature generalized thermoelastic theory, Journal of Mechanical Science and Technology 28 (2014), 4193–4198.

Abbas

I.A.

and Abo-Dahab

, On the numerical solution of thermal shock problem for generalized magneto-thermoelasticity for an infinitely long annular cylinder with variable thermal conductivity, Journal of Computational and Theoretical Nanoscience 11 (2014), 607–618.

10.

Abbas

I.A.

, Eigenvalue approach on fractional order theory of thermoelastic diffusion problem for an infinite elastic medium with a spherical cavity, Applied Mathematical Modelling 39 (2015), 6196–6206.

11.

Abbas

I.A.

and Alzahrani

F.S.

, Analytical solution of a two-dimensional thermoelastic problem subjected to laser pulse, Steel and Composite Structures 21 (2016), 791–803.

12.

Hetnarski

R.B.

Eslami

M.R.

and Gladwell

, Thermal stresses: Advanced theory and applications, Solid Mechanics and Its Applications 158 (2009), 341–369.

13.

Youssef

, Theory of two-temperature-generalized thermoelasticity, IMA Journal of Applied Mathematics 71 (2006), 383–390.

14.

Youssef

, State-space approach on generalized thermoelasticity for an infinite material with a spherical cavity and variable thermal conductivity subjected to ramp-type heating, Canadian Appl Math Quarterly 13 (2005), 369–390.

15.

Youssef

H.M.

, Two-dimensional generalized thermoelasticity problem for a half-space subjected to ramp-type heating, European Journal of Mechanics-A/Solids 25 (2006), 745–763.

16.

Youssef

H.M.

, Two-dimensional problem of a two-temperature generalized thermoelastic half-space subjected to ramp-type heating, Computational Mathematics and Modeling 19 (2008), 201–216.

17.

Jumarie

, Laplace’s transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative, Applied Mathematics Letters 22 (2009), 1659–1664.

18.

Tzou

D.Y.

, A unified field approach for heat conduction from macro-to micro-scales, Transactions-American Society of Mechanical Engineers Journal of Heat Transfer 117 (1995), 8–16.

19.

Youssef

, A two-temperature generalized thermoelastic medium subjected to a moving heat source and ramp-type heating: A state-space approach, Journal of Mechanics of Materials and Structures 4 (2010), 1637–1649.