Thermal- Elastic Waves of Microstretch Semiconductor Medium with Rotation Field during Photothermal Excitation

: A mathematical novel model for elastic semiconductor medium with microstretch properties is investigated. The generalized model is studied in the context of photo-thermoelasticity theory when the semiconductor medium is excited. The governing equations describe the coupled between the propagation of the elastic-thermal-plasma waves when the thermo-microstretch elastic semiconductor material is studied during a rotation field. The linear medium has an isotropic properties. The photothermal transport processes occurs during a two dimension (2D) elastic and electronic deformation when the microinertia of microelement is taken into account. The harmonic wave method can be used to obtain the general solutions for the basic physical variables. The complete analytical solutions of the considered variables are obtained when some mechanical-thermal and plasma conditions are applied on the boundary of the semiconductor medium. The numerical simulations of silicon (Si) and Germanium (Ge) media are constructed graphically with many comparisons according to a new parameters with thermal memories and rotation parameter.

The stress coefficient tensor.
The density of the sample.   The microstress (first moment) tensor

Introduction
Semiconductor materials have great importance and uses in modern industry. As these materials do not conduct electricity under normal conditions and have resistance to the internal movement of electrons. But when the semiconductor medium exposed to some external influences such as a change in temperature and resistance decreases, its internal properties also change. When the change in body temperature is not only the result of exposure of the medium to external and internal heat sources, but also deformations of the process itself during the microinertia process of the microelements. These changes occur during two processes, the thermo-elastic and the electron deformation (TD and ED respectively). During these processes may be prescribed the mechanical, thermal, plasma waves distribution. Plasma waves appear as a result of excited electrons on the surface of the semiconducting material which move freely in what is known as carrier density. On the other hand, thermal and mechanical waves occur as a result of the movement and collisions of internal particles of the elastic semiconducting material. In this case, the thermal and mechanical influences have a relation with stresses and strains that occur in elastic semiconductor body. In any case, as a result of these overlaps, the linear micropolar photo-thermoelasticity theory can be taken into consideration. In this case, the body microstructure impact is very important and this impact gives a new results of wave distributions can't be exist in classical photo-thermoelasticity theory.
The microstructure of body is studied during the thermal-elastic interaction processes of isotropic homogeneous elastic body in the context of the generalized thermoelasticity theory [1]. The microstructure impacts on the wave propagations are developed by Eringen and Şuhubi [2]. The new theory with micropolar influence is constructed in the context of the generalized thermoelasticity theory by Eringen [3]. Recently, the new theory is called microstretch thermoelasticity theory 4 which it appears as a special case of theory of micromorphic. After that many authors developed the generalized microstretch thermoelasticity theory with micropolar when they studied the reflection and refraction waves with two temperatures theory during a liquid adjacent to an elastic thermo-microstretch medium [4][5][6]. The linear generalized thermo-microstretch of elastic media studied the heat conduction as a thermal waves with a finite speed [7]. Lotfy and Othman et al. [8,9] used the effect of external gravitational field, magnetic field and hydrostatic initial stress with the thermal memories influences during the studying of the generalized -thermo-microstretch theory. Many applications of thermomicrostretch theory during a hydromechanics viscoelastic porous media are used with the thermal radiation effect [10,11]. On the other hand, Ezzat et al. [12,13] used the state space approach to study the viscoelastic fluid porous medium flow under the effect of magnetic field and perfectly conducting for boundary-layer over a stretching wall. Carrera and Valvano [14,15] studied a variable kinematic for shell and plate formulation for when used thermal stresses analysis of laminated structures. On the other hand, Cinefra et al. [16,17] investigated the thermal stress analysis and Heat conduction of laminated composites by a variable kinematic shell element. When a laser beams or light fall on an intracavity spherical semiconductor medium, the photothermal phenomenon is appeared [8]. The spectroscopy of photoacoustic with a sensitive analysis is used obtain the real measuring of thermal, mechanical and plasma waves of semiconducting material to [9]. Many applications in mechanical and electrical engineering used ultrasensitive laser spectroscopy with photothermal transport process [20][21][22]. Recently, the photothermal effect of semiconductor elastic materials during 2D thermal and electronic deformation is studied [23]. The generalized thermoelastic vibrations during the electronic deformation mechanism are studied in the context of the 5 optically excited to discuss the spectroscopy of photoacoustic analysis [24,25].
Lotfy et al. [26][27][28][29][30][31][32] developed many physical problems in a new photothermoelasticity theory when they studied a different applications in modern physics. A dual phase-lags models are used to study photothermal excitation during the interaction processes of a semiconductor materials [33,34]. After that, the memory responses of the photo-thermoelasticity theory when the physical properties of the semiconductor elastic medium depend on the change of the temperature with many external fields are studied [35][36][37][38][39]. Ezzat [40,41] investigated The hyperbolic and fractional thermal-plasma-elastic wave propagations is studied in a non-metallic semiconductor of organic medium.
In all the above investigations the optical properties and the inner structure of the semiconductor material are not taken into account. But in this work the governing equations are studied under the effect of rotation when the interaction between the microstretch (inner-structure) theory and the generalized photothermoelasticiy occur. In this case, the microinertia of microelement are taken during 2D (in the space (x, z)) thermo-elastic and electronic deformation. The main equations describe the physical variables distributions in the generalized photo-thermo-microstretch semiconductor elastic medium with various thermal memories. The harmonic wave with normal mode technique is used when a different mechanical-thermal-plasma conditions are applied at the free surface.
Some algebraic techniques with numerical calculations are used to obtain the complete main solutions of the physical quantities. The obtained results are shown with some comparisons graphically and discussed.

Mathematical model and main equations
Many authors over the years constructed many models in the theory of the generalized thermoelasticity [38][39][40]. During the generalized photo-thermo- 6 microstretch semiconductor medium which is taken in a Cartesian coordinates.
The photothermal with optical mechanism (carrier charges (density) are generated and the plasma wave propagation appear ) is generated at the free surface is due to the thermal effects. The interaction processes between the plasma-thermal-elastic waves during the transmission microstretch excitation are occurred. In the case of uniform rotation field with angular velocity n    of a semiconductor medium around the axial y (where n is a unit vector in the direction of y-axis) (see the geometry of the problem). The governing equations which describe a 2D photothermo-microstretch theory under the effect of rotation can be obtained as follow: Geometry of the problem (I) The interaction between thermal distribution and carrier density (plasma wave) in the context of the microstraetch photo-thermal excitation process can be given as [23]: (1) (II) The equations of motion during the microstraetch photo-thermal transport process under the effect of a uniform rotation vector when the 7 microinertia process of the microelements is taken into account can be written as follows [42]: (2) (III) The heat conduction equation which describe the coupling between the photo-thermal-elastic-microstretch processes of semiconductor medium can be given as [19]: (IV) The constitutive relations for the elastic microstretch semiconductor material during the photo-thermoelasticity theory which can be given as: (V) The tensor form of the constitutive equations for generalized photothermo-microstretch theory can be obtained as [43]: In the context of a 2D deformation, the displacement vector u  , the scalar microstretch function and rotation vector can be analyzed in xz-plane as follows: The 2D strain relation in terms of the displacement components can be given as: . In 2D deformations, the governing equations (2)-(5) can be rewritten as: According to the photo-thermal-microstretch theories, the thermal relaxation times and the parameters The following dimensionless form can be introduced for more simplification: The non-dimensional equation (16) can be used to simplify the governing equations (with dropping the prim), yields: The potential scalar functions as , can be introduced to simplify the above field equations. The dimensionless of the displacement vector can be presented in terms of the potential scalar and vector functions according to Helmholtz's theory as follows: Therefore, equation (23) can be used of the main equations (18)- (22) for more simplification, in this case equations (17)-(22) can be simplified as: Here,

3.Harmonic wave analysis
To evaluate the analytical solutions in 2D deformation of the main physical quantities which propagated in the plane wave, in this case the physical fields can be constructed by the harmonic waves (normal mode technique) method as [40][41][42]: However, the quantities represent the amplitude values of the main field quantities, , ω is frequency of complex time and b expresses the wave number in the direction of z-axis. Using equation (30) to convert the partial differential equations (17) and (24)-(28) to ordinary for equations as follow: Where, According to the linearity property, the solutions of the above ODE equation (38) can be written in the following form: On the other hand, the quantities The relations between n  ,obtained can be parameters n from the basic equations (31)-(36) as follows: On the other hand, the displacement components in terms of the unknown parameters n  can be obtained from equations (23), (30) and (47), which can be rewritten in the following form:

4.Boundary conditions
In this section, some conditions are applied at the free surface of the microstretch photo-thermoelastic semiconductor medium (the vertical plan) to obtain the values of the undetermined parameters n  . The mechanical loads with isolated thermally condition and microstretch photo-thermoelastic conditions in the context of a recombination plasma process at the free surface can be chosen as [43]: Applying the boundary conditions equation (57)

1. The theory of generalized microstretch-thermoelasticity
When carrier density t) r N , (  is neglected (i.e. 0  N ) in the main equations, the problem is discussed in the generalized microstretch thermoelasticity theory only [22].

2. The generalized photo-thermoelasticity theory
The generalized photo-thermoelasticity theory is studied only when the microstretch parameters are ignored. In this case, the governing equations are written in the generalized photo-thermal-elastic for semiconductor medium without stretch [35].

3. Different theories of the microstretch photo-thermoelasticity
The effect of thermal relaxation times can be obtained during the governing equations of microstretch photo-thermoelasticity processes according to the following relations [44]:

Discussion and numerical results
To carry out the numerical simulations for this work the semiconductor materials for example silicone (Si) and germanium (Ge) are used with helping the Matlab computer program. The physical (elastic, optical and thermal) constants of Si and Ge are used to discuss the wave propagation of the main physical fields.

Conclusion
The Many application can be used when the interaction between the thermal, plasma, microstretch and mechanical fields in elastic semicondactors are occurred. That 18 applications can be obtained in modern aeronautics, astronautics, mechanical engineering and nuclear reactors.

Appendix
The basic coefficients of equation (38) are: ( )

B b A a a b A A b A A b A a a b A A A a A A A A a b A A A A A A A A
)( )) ( ) (( ) ) ( )

A A A A A A a b A A A a b A A A A A A A A A A A A A A c A A A
On the other hand, the basic coefficients of equation (47)    (63) The main parameters of equations (52) can be obtained as:

Figure 1
The variation of the main fields against the horizontal distance of Si medium under the influence of three different theories according to the thermal relaxation times.

Figure 2
The comparison between Si and Ge materials of the main physical fields against distance in the generalized GL theory.
28 Figure 3 The comparison between the main physical fields against the horizontal distance under the effect of rotation parameters in the generalized GL theory of Si medium.
29 Figure 4 The comparison between the main physical fields against the horizontal distance under the effect of microstretch parameters in the generalized GL theory of Si medium. Figure 1 The variation of the main elds against the horizontal distance of Si medium under the in uence of three different theories according to the thermal relaxation times. The comparison between Si and Ge materials of the main physical elds against distance in the generalized GL theory. The comparison between the main physical elds against the horizontal distance under the effect of rotation parameters in the generalized GL theory of Si medium. The comparison between the main physical elds against the horizontal distance under the effect of microstretch parameters in the generalized GL theory of Si medium.