The Effect of Hyperbolic Two-temperatures Model on Waves Propagation in a Semi-conductor Medium Containing Spherical Cavities

This article is interested in the study of the carrier density, the redial displacement, the conductive temperature, thermodynamic temperature and the stresses in a semi-conductor material containing a spherical hole. This investigation deals with the photo-thermo-elastic interactions in a semi-conductor medium in the context of the new hyperbolic two-temperatures model with one relaxation time. The Laplace transform technique are used to obtain the problem analytical solution by the eigenvalues methods and the inversions of the Laplace transform were performed numerically. Numerical results for semi-conductor materials are shown graphically and discussed.


Introduction
The theory of thermoelasticity, which is the most common engineered structural material, plays an important role in steel stress analysis and applied mechanical science. It can describe the solid mechanical behavior of some common elastic materials like coal, concrete and wood. However, it cannot describe the mechanical behavior of many synthetic materials of polymer and clastomer type such as polyethylene. The temperature increment of body is not only caused by internal and external heat sources, but also by deformations of itself process during the micro-inertia of the microelement. In the first half of the last century, many authors used the theory of generalized thermoelasticity to describe elastic and thermal waves in elastic materials such as semiconductors (semi-insulating). In this case, semiconductor materials have been studied as an elastic support only. But at the end of the last century, various scientists studied semiconductor materials in particular their internal structures during microelectronic processes.
Biot [1] developed the coupled thermoelasticity theory (CD theory) when motived the law of Fourier heat conduction that became appropriate for modern engineering applications spicily in high temperature case. But in low temperature case, the thermoelastic models are physically unacceptable and cannot obtain equilibrium state. Lord et al. [2] (LS) inserted one relaxation time in the heat conduction equation (Fourier's law of heat conduction) to overcome this contradiction.
The thermo-elasticity model with classical two-temperatures are presented by and Chen et al. [3], temperature (the conductivity temperature * and the thermodynamically temperature * ).
Recently, Youssef et al. [6] investigated a new model in generalized thermoelsticity theory when they introduced the theory of hyperbolic two-temperature. Taye et al. [7] studied the hyperbolic two-temperature semiconductor thermoelastic wave by laser pulses. Saeed and Abbas [8] studied the hyperbolic two-temperatures photothermal interaction in a semi-conductor medium. Abbas et al. [9] discussed the hyperbolic two-temperature photothermal interactions in a semi-conductor material with a cylindrical cavity. Lotfy et al. [10] investigated the effect of variable thermal conductivity of a semiconducting medium with cavities under the fractional-order magnetophotothermal model. Lotfy et al. [11] investigated the response of Thomson and magnitic impact of semiconducor material due to laser pulses under photothermoelastic theory. Hobiny and Abbas [12] investigated the photothermal interaction in a two-dimension semi-conductor plane under the GN model. Ali et al. [13] studied the reflections of wave in a rotating semi-conductor nanostructure material through torsion-free boundary conditions. Yasein [14] discussed the influences of variable thermal conductivity of semi-conductor medium under photothermal model due to thermal ramp type. Lotfy et al. [15] discussed the Thomson and electromagnetic effects under the photo-thermal model of a rotator semiconductor materials with hydrostatic initial stress. Alzahrani and Abbas [16] studied the photo-thermoelastic interaction in a semi-conductor plane without energy dissipations. Abbas and Hobiny [17] used the finite differnce method to study the photothermal interraction in simecondactor medium. Youssef and El-Bary [18] studied the characterization of the photo-thermal interactions of a semiconductor solid sphere due to the fractional deformations, the thermal relaxation times, and various references temperature under L-S model. Lotfy et al [19] the photo-thermal excitations process during hyperbolic two-temperature model for magnetothermo-elastic semiconductor material. Hobiny and Abbas [20] investigated the photo-thermal wave in an infinite semi-conductor medium containing cylindrical hole. Many authors [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36] solved several problems by using numerical and analytical approaches for thermal and elastic waves.
This work is devoted to an investigate of the analyticcal solutions of photothermal interaction in semi-conductor mediums with a spherical hole under the new hyperbolic of two-temperatures thermo-elasticity. The effect of the two-temperature parameter on the thermodynamic and the conductive temperatures, the stress, the redial displacement and the carrier density distributions have been depicted graphically.

Basic equations
In this article, theoretical dissuasions during the heat transport process when the internal structure of the semi-conductor is taken into consideration. The interactions between thermal and elastic waves of the plasma are generated in the context of the own temperature (both hyperbolic temperatures). The governing equations under photo-thermal model with the hyperbolic twotemperatures in semiconductor medium can be given by [6,37,38]: The equations of motion: (1) The coupling between thermoelastic and plasma waves can be expressed as The equation of heat conduction The new hyperbolic of two-temperature relation ̈−̈= , .
(4) The stress-strain relations are expressed as Let us consider a homogeneous, isotropic unbounded semi-conductor medium containing a sphherical hole, whose state can be expressed in terms of the space variable and the time which occupying the region ≤ < ∞. The radial displacement = ( , ) non-vanishing only due to symmetry, hence the equations (1)-(5) can be rewritten by: with

Application
The initial conditions are supposed to be homogeneous. The bounding internal surface of cavity have the boundary conditions by the following To get main fields in dimensionless form, the following non-dimension variables can be used where = and 2 = +2 ρ .

Laplace transform
For G( , ) function Laplace transform is defined by Hence, the governing equations can be rewritten by Differentiating equations (24), (25) and (26) with respect to and using equation (27), yields: The eigenvalues of matrix are the three roots of equation (35) The solutions of equations (34) which are bounded as → ∞ can be given by where 3/2 is the modified of Bessel's function of order

Numerical Results and Discussions
To theoretically study the results obtained, the physical properties and physical constants of

Availability of Data and Material
There is no data or material that has been copied from elsewhere in the proposed manuscript.   Fig. 1 The carrier density variations versus the redial distance.     Clas. Two-Temp. Hyper. Two-Temp.   Clas. Two-Temp. Hyper. Two-Temp.