The Fractional Strain Influence on a Solid Sphere under Hyperbolic Two-Temperature Generalized Thermoelasticity Theory by Using Diagonalization Method

Mathematics Department, Faculty of Education, Alexandria University, Alexandria, Egypt Mechanical Engineering Department, College of Engineering and Islamic Architecture, Umm Al-Qura University, Makkah, Saudi Arabia Basic and Applied Science Institute, Arab Academy for Science, Technology, and Maritime Transport, P. O. Box 1029, Alexandria, Egypt Mathematics Department, Al-Lith University College, Umm Al-Qura University, Al-Lith, Saudi Arabia


Introduction
e subject matter's primary concern is how to detect an exact model that simulates thermoelastic material's behavior. Researchers and authors have developed several models in which the waves in solids and thermoelastic materials have been transmitted. But not all these models succeed as one of the metrics of a good model is to give the experimental results with a finite speed of progress of mechanical and thermal waves. To talk about thermomechanical transition models with elastic materials, a wide area cannot be limited to study. erefore, we will discuss some recent models that need to be addressed. e thermoelasticity model has been developed by Chen and Gurtin, based on two different conductive and dynamic temperatures. e temperature differential is proportionate to the heat source [1]. In the two-temperature thermoelastic theory, Warren and Chen investigated wave propagation [2]. However, before Youssef updates this theory and creates a two-temperature generalized thermostat model, there will be no study of that theory [3]. In many applications and inquiries, Youssef and several other writers used this model [4][5][6][7][8][9][10][11][12][13]. Youssef and El-Bary have proven that the classical two-temperature generalized thermoelasticity model does not provide heat wave transmission at a finite speed [14]. erefore, Youssef and El-Bary have changed this model and implemented a new two-temperature model based on different thermal conductivity rules, called hyperbolic twotemperature generalized thermoelasticity [14]. In that model, Youssef suggested that it is proportional to the heat supply to differentiate between conductive temperature and dynamic temperature acceleration. e speed of thermal wave propagation is limited in this model. e two-temperature theory is considered as a type of generalization of the heat conduction equation. It helps us to separate the conductive thermal wave and dynamical thermal wave and study each wave and its effects on the materials. e fact that fractional systems possess memory justifies this generalization, as the time evolution of romantic relationships is naturally impacted by memory [15,16]. e fractional-order dynamic model could model various real materials more adequately than integer-order ones and provide a more adequate description of many actual dynamical processes [17,18]. us, some new thermoelastic models based on the concept of fractional calculus were introduced. Magin and Royston developed the first model using the fractional deformation derivative characterizing the material's behavior [19]. A Hookean solid is the derivative's zero-order, while a Newtonian fluid is the oneorder. e intermediate spectrum of the heat gives different orders of thermoviscoelastic substances [19].
Most of the authors who regarded their applications as the spherical medium faced the singularity problem in its center. Few authors, for example, ibault and others, were able to solve this situation in the thermoelastic solid sphere using L'Hopital's rule to resolve this problem [36].
In the present investigation, a solid sphere in the context of hyperbolic two-temperature generalized thermoelasticity theory based on the fractional-order strain definition will be studied. e advantages of the current article are that it is considered more generalized model than the models that precede it and contains the results of a new theory with comparing it to the results of previous theories.

The Governing Equations
Consider a perfect, thermoelastic, conducting, and isotropic spherical body that occupies the region ξ � (r, ψ, ϕ) : 0 ≤ r ≤ a, 0 ≤ ψ ≤ 2π, 0 ≤ ϕ < 2π}. We use a spherical coordinative system (r, ψ, ϕ) that displays the radial coordinate, colatitude, and longitude of a spherical system, without any forces on the body and initially calming where r is the sphere radius, as in Figure 1. When there is no latitude or longitudinal variance, the symmetry condition is fulfilled. Both state functions depend on the distance and time of the radius.
We note that, due to spherical symmetry, the displacement components have the form e equation of motion [10,20] is e constitutive equations with damage mechanics variable [10,20] are where e is the cubical dilatation and it is given by e hyperbolic two-temperature heat conduction equations take the forms [10,14,20] where c(m/s) is the hyperbolic two-temperature parameter [14] and ∇ 2 � (1/r 2 )(z/zr)(r 2 (z/zr)). e Riemann-Liouville fractional integral I α f(t) description is used in the above equations written in a convolution type form [20,37] (11) which provides Caputo with the form of fractional derivatives: We consider that φ � (T C − T 0 ) and θ � (T D − T 0 ) are the conductive and dynamical temperature increments, respectively. en equations (2)-(5), (9), and (10) take the forms Equation (13) can be rewritten to be in the form e following nondimensional variables are used for convenience [5,9,10]: en, we obtain o . e primes are suppressed for simplicity. e operator ∇ 2 � (1/r 2 )(z/zr)(r 2 (z/zr)) is singular at r � 0; however, if symmetry prevails, the singularity will be reduced by the following L'Hopital's rule [36]: e Laplace transform of the fractional derivative is defined as [37] ℓ e following initial conditions have been assumed: us, equations (19)- (24) have the forms where δ 2 � 3c 2 /s 2 . Substituting equation (35) into equations (33) and (34), we get where Substituting equation (41) into equation (40), we obtain where

The Diagonalization Method
We can rewrite equations (41) and (42) in a matrix form as follows [38]: For simplicity, we write the system in (43) as a homogenous system of linear first-order differential equation as [38] where Matrix A has four linearly independent eigenvectors; hence, we can construct a matrix V from the eigenvectors of If we make the substitution Z � VY in system (44), then which gives where ±k 1 and ± k 2 are the eigenvalues of matrix A or the roots of the characteristic equation where Since W is a diagonal matrix, then system (46) is uncoupled, making each differential equation in the system have the form y i ′ � k i y i , i � 1, 2, 3, 4. e solution to each of these linear equations is y i � c i e k i x , i � 1, 2, 3, 4. Hence, the general solution of system (46) can be written as the column vector [38]: en, the final solution of system (44) is Matrix V from the eigenvectors of matrix A takes the form Substituting equations (49) and (51) into equation (50), we get e boundary conditions in (27) and equation (52) give that Hence, we obtain To get the constants A 1 and A 2 , we must apply the boundary conditions at r � a; we consider the sphere when r � a is thermally shocked as follows: where H(t) is the Heaviside unit step function and φ o is constant.
For the mechanical boundary condition, we consider the surface of the sphere when r � a is connected to a rigid foundation, which can stop any displacement (u| r�a � 0), offering zero volumetric deformation from equation (39); hence, we have e(a, t) � 0. (57) Applying Laplace transform on equations (56) and (57), we obtain e(a, s) � 0. (58) Applying the boundary conditions in equations (54) and (55), we get the following system: By solving system (59) and (60) by using the relations between the roots (48), we get Hence, we have To obtain the displacement function, we will use equations (39) and (63) We use the average of the three principal stress components for equations (36)- (38) to achieve a straightforward way of distribution to be as follows: e techniques of Riemann-sum approximation are used to obtain numerical solutions of studied functions in the time domain. In this method, any function can be transformed into the Laplace by using the following formula [39]: where "i" is the well-known unit of imaginary number and "Re" gives the real part. Many computational studies have shown that value κ meets the relationship for convergence with faster procedures. κt ≈ 4.7 [39]. After calculating the Laplace transformations, the stressstrain energy can be obtained in the following formula [40]: For the current model, the stress-strain energy is in the form ϖ(r, t) � 1 2 σ rr e rr + σ ψψ e ψψ + σ ϕϕ e ϕϕ .
After eliminating the term with a small value, we get

Numerical Results and Discussion
For numerical analysis, copper is the thermoelastic substance for which the various physical constants' values are taken as follows [24]: (72) us, we get the nondimensional values of the problem as b � 0.01047, For a large range of the dimensionless radial distance, r (0 ≤ r ≤ 8.0), the numerical results from conductive and dynamic temperature increments, pressure, changes, average stress, and stress-power distributions were shown at the instant value of dimensionless time t � 1.0. Figures 2-7 were obtained for various values of the twotemperature parameter c � (0.0, 0.5), which gives δ 2 � 3c 2 /s 2 � (0.0, 3(0.5) 2 , 3(0.5/s) 2 ), where the value δ � 0.0 represents the L-S one-temperature model; it has been figured in solid curves. e value c � 0.5 represents two cases; the first case is δ 2 � 3(0.5) 2 , which represents the classical two-temperature model and has been figured with dash curves, while the second case is δ 2 � 3(0.5/s) 2 , which represents the hyperbolic two-temperature model and has been figured in dot curves. e numerical results of those figures have been calculated when the fractional order parameter α � 0.5. Figure 2 shows the conductive temperature increment distributions, and it is seen that all the curves start from the position r � 8.0 with the value φ(r � 8.0) � 1.0 , which is the value of the thermal shock on the bounding surface of the sphere. All the curves have the same behavior but with different values. e two-temperature parameter has a significant effect on conductive temperature distribution. At the endpoint of the sphere's radius, the one-temperature and hyperbolic two-temperature model curves fall to zero values. It means that the conductive thermal wave has propagated with a finite speed in the context of one-temperature and   One-temp. Class. two-temp. Hyp. two-temp Mathematical Problems in Engineering hyperbolic two-temperature models, while the classical twotemperature model does not. Figure 3 shows the dynamical temperature increment distributions, where it is noted that all the curves start from the position r � 8.0 with different values θ| one−temp. � 1.0, which is the value of the thermal shock on the bounding surface of the sphere θ| Class. two−temp. � 0.98 and θ| hyp. two−temp. � 0.75. All the curves have the same behavior with various values. e two-temperature parameter has a significant effect on the dynamical temperature distribution. At the endpoint of the sphere's radius, the one-temperature and hyperbolic two-temperature model curves fall to zero values. It means that the dynamical thermal wave has propagated with a finite speed in the context of one-temperature and hyperbolic two-temperature models, while the classical two-temperature model does not. e volumetric distribution of the strain is illustrated in Figure 4, and three curves begin at r � 8.0, a position with zero value e(r � 8.0) � 0.0 , which agrees with the boundary condition on the sphere's bounding surface. All the curves have the same attitude with different values. e onetemperature and hyperbolic two-temperature models' curves have a sharp point in the same position with different values. In contrast, the classical two-temperature model has a peak point in different situations. e absolute values of the maximum volumetric deformation take the following order: (74) Figure 5 shows the displacement distributions, and it is seen that the three curves start from the position r � 8.0 with zero value u(r � 8.0) � 0.0. All the curves have the same attitude with various values. e one-temperature and hyperbolic two-temperature models' curves have sharp points in the same position with different values. In contrast, the classical two-temperature model has a peak point in different situations. e absolute values of the maximum displacement take the following order: (75) Figure 6 represents the average stress distributions, and we can see that the three curves start from the position r � 8.0 with different values. e one-and two-temperature curves have the same behavior with different values, and each has a sharp point. e curve, by comparison, is distinct and smooth in the classical two-temperature model. Figure 7 shows the stress-strain energy distribution, and it is noted that the three curves start from the position r � 8.0 with zero values. e three curves have the same behavior with different values, and each one has a peak point. e maximum values of the stress-strain energy take the following order: Figures 8-13 have been done for different values of the fractional-order parameter α � (0.0(τ 1 � 0), 0.2, 0.6, 1.0) in the context of the hyperbolic two-temperature model to discuss its effects on the state of the studied functions. e case α � τ 1 � 0.0 represents the normal stress-strain relations as the usual Hooke's law of the elastic body. In contrast, the cases α � (0.2, 0.6) define the new cases between the normal elasticity and the viscoelasticity and α � 1.0 represents the viscoelasticity. Figures 8 and 9 show that the fractional-order parameter has limited impacts on the dynamical and conductive One-temp. Class. two-temp.
Hyp. two-temp temperature increment. is result was expected because the fractional-order parameter's effect exists firmly in the stressstrain relation, which significantly influences the mechanical waves more than the thermal waves. Figure 10 represents that the effect of the fractionalorder parameter is significant on the volumetric strain distributions. All the curves start from the position r � 8.0 with zero value, and every curve contains a peak except for the curve, which has no fraction τ 1 � 0.0, and it has a sharp point.
e maximum values of the absolute value of the volumetric deformation take the following order: (77) Figure 11 indicates a significant influence on displacement distributions of the fractional-order parameter. e four curves begin at the location r � 8.0, and each curve contains a peak except the curve, which does not have a fraction τ 1 � 0.0, where it has a sharp point. e maximum values of the absolute value of the displacement take the following order:  Figure 12 represents that the fractional-order parameter has significant impacts on stress distributions. All the curves start from the position r � 8.0 with the same value, and each curve is smooth except the curve of the case, which has no fraction τ 1 � 0.0, and it has a sharp point. Figure 13 represents that the fractional-order parameter has significant effects on stress-strain energy distributions. e four curves start from the position r � 8.0 with zero values, and each curve has a peak point. e values of the maximum stress-strain energy take the following order: ϖ max (α � 1.0) > ϖ max (α � 0.6) > ϖ max (α � 0.2) > ϖ max α � τ 1 � 0.0 .
For the validation of the results, we can notice that the current results that are based on traditional integer derivative agree with the results in the past publications [25,28,29,32,[41][42][43].

Conclusions
e numerical results conclude that the one-temperature model and the hyperbolic two-temperature theory of thermoelasticity generate thermomechanical waves that can propagate with limited speeds. erefore, the hyperbolic two-temperature thermoelasticity model is a successful model to describe thermoelastic materials' thermodynamical behavior. Moreover, the two-temperature parameter has significant impacts on the states of all the studied functions. e fractional-order strain parameter has weak effects on the thermal waves, while its effects on the mechanical waves are significant. � ((λ + 2μ)/μ) 1/2 c: � (3λ + 2μ)α T ε:

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.