Fractional-Order Thermoelastic Wave Assessment in a Two-Dimensional Fiber-Reinforced Anisotropic Material

: The present work is aimed at studying the e ﬀ ect of fractional order and thermal relaxation time on an unbounded ﬁber-reinforced medium. In the context of generalized thermoelasticity theory, the fractional time derivative and the thermal relaxation times are employed to study the thermophysical quantities. The techniques of Fourier and Laplace transformations are used to present the problem exact solutions in the transformed domain by the eigenvalue approach. The inversions of the Fourier-Laplace transforms hold analytical and numerically. The numerical outcomes for the ﬁber-reinforced material are presented and graphically depicted. A comparison of the results for di ﬀ erent theories under the fractional time derivative is presented. The properties of the ﬁber-reinforced material with the fractional derivative act to reduce the magnitudes of the variables considered, which can be signiﬁcant in some practical applications and can be easily considered and accurately evaluated.


Introduction
Fiber-reinforced composites are used widely in structural engineering. Continuous models are used to illustrate the mechanical properties of these materials. Fibers are supposed to have inherent material properties, rather than some form of inclusions in the model as in Spencer [1]. A fiber-reinforced thermoelastic material is a composite material that exhibits strongly anisotropic elastic behaviors such that elastic parameters have extensions in the fiber directions that are on the order of 50 or more times greater than their parameter in the transverse directions. This composite material is lightweight and has high strength and rigidity at high temperatures. Due to practical and theoretical importance, several problems with wave and vibration in fiber-reinforced mediums have been studied. The idea of introducing a continuous self-reinforced for every point of an elastic solid was presented by Belfied et al. [2]. The models were then applied to the rotations of a tube by Verma and Rana [3]. Verma [4] also studied the magneto-elastic shear wave in self-reinforcing media. Sengupta and Nath [5] studied the problem of surface waves in fiber-reinforced anisotropic elastic materials. Hashin and Rosen [6] discussed the elastic modulus for fiber-reinforced mediums. Singh and Singh [7] discussed the problem of reflections of a plane wave on the free surfaces of a fiber-reinforced elastic plane. Chattopadhyay and Choudhury [8] investigated the problem of propagations, reflections, and transmissions of magnetoelastic shear waves in self-reinforcing media.

Mathematical Model
The basic equations in the context of generalized fractional thermoelastic theory with one relaxation time for an anisotropic fiber-reinforced medium in the absence of a body force and heat source are given by: σ ij = λe kk δ ij + 2µ T e ij + α a k a m e km δ ij + a i a j e kk + 2(µ L − µ T ) a i a k e k j + a j a k e ki + βa k a m e km a i a j − γ ij ( By taking into consideration the above definition, it can be expressed by where I υ is the Riemann-Liouville integral fraction introduced as a natural generalization of the well-known integral I υ g(r, t) that can be written as a convolution type, where g(r, t) is Lebesgue's integral function and Γ(υ) is the Gamma function. In the case where g(r, t) is definitely continuous, it is possible to write We consider plane waves in the xy-plane; therefore, in the two-dimensional fiber-reinforced medium, we have written The fiber direction is chosen such that a = (1, 0, 0) so that the preferred direction is the x-axis and Equations (1)-(3) can be expressed by with where 11 , γ 22 = (2λ + α)α 11 + (λ + 2µ T )α 22 , and α 11 , α 22 are the linear thermal expansion coefficients.

Initial and Boundary Conditions
The initial conditions of the problem are given as while the problem adequate boundary conditions are expressed as For suitability, the nondimensionality of the physical quantities can be taken as where η = ρc e k and c = c 11 ρ . In these nondimensional terms of parameters in Equation (16), the basic Equations (8)- (15) can be written as (after ignoring the superscript ' for appropriateness) with σ yy = f 4 ∂v ∂y where ρc e , and f 8 =

Method of Solution
Now, we can apply Laplace transforms, which defined by where the Laplace transforms parameter is s, whereas the Fourier transforms for any functions h(x, y, s) can be expressed as Thus, the governing equations with the boundary conditions under the initial conditions are presented to obtain the ordinary differential equations as follows: − with Now, we obtain the general solutions of Equations (26)-(28) by the eigenvalues method proposed [36,[43][44][45][46]. From Equations (26)-(28), the vectors matrix can be expressed by where The characteristic equation of matrix A takes the form where R 1 = a 41 a 53 a 62 − a 63 a 52 a 41 , R 2 = a 52 a 41 − a 62 a 53 − a 62 a 54 a 46 + a 63 a 41 + a 63 a 52 + a 63 a 54 a 45 + a 64 a 52 a 46 − a 64 a 53 a 45 , The roots of the characteristic Equation (34), which are also the eigenvalues of matrix A. In the cases where ξ 1 , −ξ 1 , ξ 2 , −ξ 2 , ξ 3 , and −ξ 3 are the eigenvalues, the conforming eigenvectors of eigenvalues ξ can be calculated as The solutions of Equations (33) can be given by where the terms containing exponentials of growing nature in the space variable Z have been discarded due to the regularity condition of the solution at infinity, and A 1 , A 2 , and A 3 are constants to be determined from the boundary condition of the problem. Now, for any function h * (x, q, s), the transforms of the Fourier inversion are given by Finally, to obtain the general solutions of the variations in temperature, the components of displacement, and the components of stresses with respect to the distances x, y for any time t, the Stehfest [35] numerical inversion method is used. In this method, the inverse of Laplace transforms for h(x, y, s) can be expressed as where where N is the number of terms.

Numerical Result and Discussion
In order to illustrate the theoretical outcomes obtained in the preceding sections, we give some numerical values for the physical parameters [47]: The field quantities, the increment in temperature T, the displacement components u, v, and the components of stresses σ xx , σ xy depend not only on space x, y and time t, but also on the fractional order of the time derivative ν. The numerical computations for all the nondimensional field quantities of the plate are demonstrated in Figures 1-13. Figures 1-3 show the temperature contours on the plate for different values of the fractional-order parameter ν when t = 0.6. We can find that the temperature-change zone is restricted in a finite area and the temperature does not change out of this area. The white color in these figures refers to a temperature variation of zero in this region. We can observe that the region with changes in temperature become larger with the generalized thermoelastic theory without fractional time derivative ν = 1, while out of this region, the temperature maintains the original values. Figure 4 displays the variation in temperature with respect to the distance x, and it indicates that the temperature field has maximum values at the boundary and, after that, decreases to zero. Figure 5 shows the variations in horizontal displacement along the distance x. It is apparent that when the surface of the half-space is taken to be traction-free, and the heat flux is applied on the surface, the displacement for various values of fractional order parameter ν shows a negative value at the boundary of the half space. In addition, it attains stationary maximum values after some distances and then decreases to zero. Figure 6 shows the variations in vertical displacement with respect to x for various values of fractional order parameter ν, where we observed that a significant difference in the value of displacement is noticed for the different values of ν. Figures 7 and 8 display the distributions of stress components σ xx , σ xy with respect to the distance x for different values of fractional order parameter ν. It is observed that the components of stress σ xx , σ xy always start from the zero value and terminates at the zero value to obey the problem boundary conditions. Figure 9 shows the temperature variations T along the distance y and indicates that the variations in temperature have maximum values at the length of the heating surface y ≤ 0.5 , and start to reduce completely close to the edges y ≤ 0.5 where they reduce smoothly and, in the end, reach zero. Figures 10 and 12 display the horizontal displacement variations u and the stress component σ xx with respect to the distance y. They indicate that they have maximum values at the length of the thermal surface y ≤ 0.5 , and they begin to decrease completely close to the edges (y = ± 0.5) and, after that, reduce to zero. Figures 11 and 13 display the vertical displacement variations v and the stress component σ xy with respect to the distance y. It is noticed that they begin increase and reach maximum values just near the edges (y = ± 0.5), and then decreases close to zero after that.
As expected, it can be found that the fractional parameter has great effects on the values of all the physical quantities. According to the numerical results, this new fractional parameter of the generalized thermoelastic model offers finite speed of the thermal wave and mechanical wave propagation.

Conclusions
In this article, we have studied the solutions of a two-dimensional problem for an infinite fiberreinforced thermoelastic material. Based on the eigenvalue scheme, Laplace transforms, and exponential Fourier transforms, the analytical solutions have been obtained. The properties of the fiber-reinforced medium with the fractional derivative act to reduce the magnitudes of the variables considered, which can be significant in some practical applications, and can be easily considered and accurately evaluated.

Conclusions
In this article, we have studied the solutions of a two-dimensional problem for an infinite fiber-reinforced thermoelastic material. Based on the eigenvalue scheme, Laplace transforms, and exponential Fourier transforms, the analytical solutions have been obtained. The properties of the fiber-reinforced medium with the fractional derivative act to reduce the magnitudes of the variables considered, which can be significant in some practical applications, and can be easily considered and accurately evaluated.
Author Contributions: The two authors conceived the framework and structured the whole manuscript, checked the results, and completed the revision of the paper. The authors have equally contributed to the elaboration of this manuscript. All authors have read and agreed to the published version of the manuscript. Acknowledgments: The authors acknowledge with thanks the University technical and financial support.

Conflicts of Interest:
The authors declare no conflict of interest.

Nomenclature u i
are the components of displacement, T, is the increment in temperature, ρ is the medium density, T o is the reference temperature, λ, µ T are the elastic constants, σ ij are the components of stresses, c e is the specific heat at constant strain, α, β, (µ L − µ T ) are the reinforced anisotropic elastic parameters, a i are the components of vector a where a 2 1 + a 2 2 + a 2 3 = 1, K jj is the thermal conductivity, τ o is the thermal relaxation time, υ is the fractional parameter, where 0 < υ ≤ 1 cover two types of conductivity, υ = 1 for normal conductivity, and 0 < υ < 1 for low conductivity q o is a constant H is the Heaviside unit function t p is the pulse heat flux characteristic time i, j, k = 1, 2, 3, are the number of components