Abstract
The present research paper illustrates how noninteger derivative order analysis affects the reflection of partial thermal expansion waves under the generalized theory of plane harmonic wave reflection from a semivacuum elastic solid material with both gravity and magnetic field in the three-phase lag model (3PHL). The main goal for this study is investigating the fractional order impact and the applications related to the orders, especially in biology, medicine, and bioinformatics, besides the integer order considering an external effect, such as electromagnetic, gravity, and phase lags in a microstretch medium. The problem fractional form was formulated, and the boundary conditions were applied. The results were displayed graphically, considering the 3PHL model with magnetic field, gravity, and relaxation time. These findings were an explicit comparison of the effect of the plane wave reflection amplitude with integer derivative order analysis and noninteger derivative order analysis. The fractional order was compared to the correspondence integer order that indicated to the difference between them and agreement with the applications in biology, medicine, and other related topics. This phenomenon has more applications in relation to the biology and biomathematics problems.
1. Introduction
Recently, due attention has been made to study the interaction between electromagnetic field, thermal field, gravity, and the influence of microstretch due to its utilitarian features within different domains, such as physics, geophysics, geology, astronomy, and engineering. Choudhuri [1] proposed the problem of thermoelasticity theory with 3PHL. Kumar et al. [2] explored the plane strain problem within microstretch elastic solid. Kumar and Partap [3] examined the reflection of plane waves in a heat flux-dependent microstretch thermoelastic solid half space. Lord and Shulman [4] established generalized theories in which they replaced the Fourier law of thermal conductivity by Maxwell-Cattaneo law introducing a thermal relaxation time parameter within Fourier’s law. Green and Lindsay [5] illustrated binary relaxation parameters at the constitutive relationships regarding the entropy and stress tensor. After some years, Green and Naghdi [6–8] introduced new three models known as GN-I, GN-II, and GN-III. They argued that the linearized form of model-I agrees with the classical thermoelastic one. In model-II, the internal production rate of is typically zero, suggesting no dissipation of thermal energy. The model discloses undamped thermoelastic waves in a thermoelastic material. It is primarily indicated as the theory of thermoelasticity in the absence of the dissipation of energy. The former models are combined in model-III as special cases. The model discloses energy dissipation generally. Abbas and Kumar [9] discussed deformation because of the heat source in micropolar generalized thermoelastic half-space using the finite element technique. The authors of [10] explored the stabile solutions concerning the 3PHL thermal conductivity. Utilizing the theory, Kar and Kanoria [11] and Banik and Kanoria [12] resolved various issues.
The initial stresses in solids considerably affect the mechanical response of the material from a primarily stressed configuration. They are widely applied in soft biological tissues’ performance, engineering structures, and geophysics. Elsagheer and Abo-Dahab [13] discussed the reflection of thermoelastic waves from insulated boundary fiber-reinforced half-space affected by the magnetic field and rotation. Abo-Dahab et al. [14] employed the Lord-Shulman and dual-phase lag models to explore how rotation and gravity affect an electro-magneto-thermoelastic medium in the presence of diffusion and voids. Using three thermoelastic theories, the authors of [15] discussed SV-waves occurrence at the interface between solid-liquid media under the initial stress and electromagnetic field. Abo-Dahab et al. [16] studied the impact of rotation and gravity on the reflection of P-waves from thermo-magneto-microstretch medium within the 3PHL model with initial stress.
Several definitions were introduced concerning fractional derivatives. For instance, Riemann–Liouville’s definition is characterized by the fractional derivative of constant is not zero and applied for the non-differentiable functions. Caputo’s definition can be applied to differentiable functions, and zero is the value of the fractional derivative of constant [17].
The fractional differential equations’ theory and applications were studied by Kilbas et al. [18]. Hilfer [17] discussed the uses of fractional calculus in physics. Katugampola [19] used a novel method to a generalized fractional integral. The authors of [20] discussed analytically solving the space-time fractional nonlinear Schrödinger equation. Abdel-Salam and Hassan [21] studied solving the class of linear and nonlinear fractional differential equations. Abdel-Salam and Yousif [22] discussed solving the nonlinear space-time fractional differential equations by the fractional Riccati expansion approach. The approximate solution to the fractional Lane-Emden kind equations was studied by Nouh and Abdel-Salam [23]. Examining the fractional derivative for natural convection in a slanted cavity having porous media was discussed by Ahmed et al. [24]. The authors of [25] discussed the impacts of the rotation, voids, magnetic field, and initial stress on plane waves in generalized thermoelasticity. Marin et al. [26] explained extending the domain of the influence theory concerning the generalized thermoelasticity of anisotropic material in the presence of voids. Saeed et al. [27] discussed the GL model on the thermoelastic interaction in a poroelastic material by the finite element approach. Abo-Dahab et al. [28] discussed the impact of the electromagnetic field in the fiber-reinforced micropolar thermoelastic medium in the context of four models.
Many authors in [29–36] studied various applications of fractional calculus in mathematical modeling with a comparison to constant cases with physics properties. The analysis of the fractional derivative order and temperature-dependent characteristics on P- and SV-waves reflection influenced by initial stress and 3PHL model were considered by Abo-Dahab et al. [37]. Alotaibi et al. [38] studied the fractional calculus of thermoelastic P-wave reflection influenced by the electromagnetic fields and gravity. Hobiny and Abbas [39] analytically solved the fractional order photo-thermoelasticity within a nonhomogenous semiconductor medium. Povstenko and Kyrylych [40] illustrated the fractional thermoelasticity issue concerning a plane with a line crack influenced by heat flux loading.
Many researchers in [41–47] investigated the effect of several variables in thermoelasticity with different models and archived all conditions to these models.
In this paper, the reflection of the plane waves from a semivacuum elastic solid material with the electromagnetic field and gravitational under the influence of relaxation times was studied. The necessary comparisons were made to simplify and explain the phenomenon at the fractional and nonfractional differentiations. The reflection appeared in the effect of the amplitude of the plane waves where the presence of the fractional differential was evident in each of the different results of the phenomenon. The fractional order was compared to the correspondence integer order that indicated the application and agreement with the applications in biology, bioinformatics, medicine, and related topics.
1.1. Formulation of the Problem
We utilized the generalized fractional thermo-microstretch’s equations in a rectangular coordinate scheme with -axis directed into the media. We take the constant magnetic field intensity to represent the -axis direction. We consider with linearized equations of electrodynamics in the presence of displacement current due to motion as (Ezzat [48])
The fractional motion equation with gravitational and Lorentz’s body forces as (Paria [49], Lord and Shulman [4]) where
Using the previous equations, the following equations are obtained:
From equations (3) and (5), we obtain
So, the displacement vector has the components
The basic governing equations become
Such that
The constitutive relations take the following form:
Moreover, we introduce these dimensionless quantities to make the solution easier
Utilizing Equation (16), the governing Equations (8)–(12) recast in this form (after suppressing the primes)
where
The components of the displacement and take the following form concerning the potential functions and
Using Equation (31) in Equations (17)–(21), we get where
1.2. Solution of the Problem
In this section, solving Equations (32)–(36) is assumed as
Substituting from Equation (38) into Equtions (32)–(36), the result is where .
To help identify the nontrivial solution, we change from Equation (39) into Equations (33)–(38) and get which tends to where , , , , , and are in the Appendix.
From Equation (45), we can identify five diverse velocities accompanying five waves.
After that, are written as
Based on Equations (40) and (41), the result is where .
Also, from Equations (42) and (43), we get
1.3. The Boundary Conditions
The problem’s boundary conditions are written as follows:
Finally,
Based on Equations (50)–(54), the result is
1.4. Numerical Results and Discussion
In this section, some numerical findings are discussed based on the illustration of the results obtained in the former sections and a comparison of such results in different cases.
Therefore, we chose these materials:
Figure 1 highlights the various values of the amplitudes concerning the angle of incidence of the reflected waves for various values of the fractional parameter of two different values of gravity . Clearly, the amplitudes of incidence waves decline when rising the gravity field. They decline then rise with the increase of to reach the unit at °.
In the case of , the amplitude of the wave is less than in the case of , indicating that the greater the effect of gravity leads to an increase in the wave’s amplitude.
Figure 2 displays the various values of amplitudes concerning the angle of incidence of the reflected waves for various values of the fractional parameter for the two diverse values of gravity that shows the oscillatory performance in the entire range of angle Clearly, the amplitudes of reflection waves rise when rising the gravity field, which rises then declines with increasing approaches zero at .
In the case of , the wave amplitude started to increase significantly than at , and the effect of the fractional differential appears more clearly in the case of or , both at or .
Figure 3 highlights the various values of amplitudes concerning the angle of incidence of the reflected waves for various values of the fractional parameter for the two values of gravity that shows the oscillatory performance in the entire range of angle Clearly, the amplitudes of reflection waves rise when rising the gravity field and decline when rising until it equals zero at .
In the case of , the effect of the wave amplitude shows a slight change in , and it is not affected by the gravity at the lowest value of the fractional differential parameter at , while the rest of the values for the fractional differential parameter are clear.
Figures 4 and 5 display the various values of amplitudes concerning the angle of incidence of the reflected waves regarding various values of the fractional parameter for the two diverse values of gravity that escalates in the whole entire of angle Clearly, the amplitudes of the reflection waves increase with a higher gravity field. They rise then decline when rising until it equals zero at . In the wave amplitude , the effect of the fractional differential parameter at takes the same behavior whether gravitational or . In the case of , the effect of the wave amplitude shows a slight change in , and it is not affected by the gravity at the lowest value of the fractional differential parameter at , while the rest of the values for the fractional differential parameter are clear.
Figure 6 illustrates the magnitude of amplitude ratios concerning the angle of incidence of the reflected waves regarding various values of the fractional parameter of two diverse values of the magnetic field . The amplitudes of incidence waves are lower when rising the magnetic field that decline then grows with rising to reach the unit at . It appears that the amplitude of the wave is lower in the case of the fractional differential parameter at along , while it increases with an increase of and is more amplitude in the case of .
Figures 7–10 display the various values of amplitudes concerning the angle of incidence of the reflected waves for various values of the fractional parameter of the two different values of the magnetic field , which escalates in the entire range of angle Obviously, the amplitude reflection waves decrease with an increase of the magnetic field and decrease with an increase of until equal zero at . In the case of amplitudes , the behavior of the waves appears to be the same in all cases for different alpha values. At the same time, the effect of the fractal calculus parameter is less amplitude than the absence of fractional differential parameter. In the case of , the impact of the magnetic field, whether or is not affected by the wave amplitude, as we see the effect of fractional differentiation in all the fractional differential parameter values.
In sum, the method of the fractional derivative technique applies to other relevant issues in geology, geophysics, physics, astronomy, and engineering, agreeing with the integer derivative if .
If the fractional order considers only an integer without an electric field, the results obtained are deduced to the results given in Othman et al. [46].
2. Conclusion
The plane harmonic waves’ reflection from a semi-infinite elastic solid of thermo-microstretch was studied in this article under the thermoelasticity theory with three-phase lag. Expressing the reflection coefficients that represent the relationships of the amplitudes of the reflected waves to the amplitude of the incidence waves obtained the reflection coefficient ratio variations with the angle of incidence with changing the dielectric constant, magnetic field, and gravity field. The results were compared in the cases of the existence and negligence of the parameter fractional.
Due to their practical issues, we recommend utilizing the research findings in other fields, such as geophysics, engineering, geology, volcanoes, earthquakes, and structures. Future works will consider taking into account the effect of radiation, rotation, and other external parameters related to the phenomenon topics applicable in the environment.
Appendix
Nomenclature
| : | Magnetic induction vector |
| : | Specific heat per unit mass |
| : | Electrical density vector |
| : | Strain tensor |
| : | Body force vector of Lorentz |
| : | Gravitational constant |
| : | Magnetic vector |
| : | Perturbed magnetic vector |
| : | Elementary constant magnetic field vector |
| : | |
| : | Microinertia moment |
| : | Electric current density vector |
| : | Thermal conduction coefficients |
| : | Coupled stress tensor |
| : | Time |
| : | The reference temperature |
| : | The temperature distribution |
| : | Displacement vector |
| : | Microstretch constants |
| : | Coefficient of linear thermal expansion |
| : | Kronecker delta |
| : | Electric permeability |
| : | Rotation vector |
| : | Scalar microstretch |
| : | Micropolar constants |
| : | Lame’ constants |
| : | Magnetic permeability |
| : | Velocity of the coupled waves |
| : | Mass density |
| : | Stress tensor |
| : | Stress tensor of Maxwell |
| : | The phase-lag of the temperature influx |
| : | The phase-lag of the heat tendency |
| : | Thermal displacement phase lag |
| : | Complex frequency |
| : | Wave number |
| : | |
| : | . |
Data Availability
The data supported from our published papers and all the results are new.
Conflicts of Interest
The authors declare no conflict of interests in publishing the present research paper.
Acknowledgments
The authors acknowledge the financial support of Taif University researchers, supporting project no. TURSP-2020/164, Taif University, Taif, Saudi Arabia.