Abstract

This article constructs a mathematical model based on fractional-order deformations for a one-dimensional, thermoelastic, homogenous, and isotropic solid sphere. In the context of the hyperbolic two-temperature generalized thermoelasticity theory, the governing equations have been established. Thermally and without deformation, the sphere’s bounding surface is shocked. The singularities of the functions examined at the center of the world were decreased by using L’Hopital’s rule. Numerical results with different parameter fractional-order values, the double temperature function, radial distance, and time have been graphically illustrated. The two-temperature parameter, radial distance, and time have significant effects on all the studied functions, and the fractional-order parameter influences only mechanical functions. In the hyperbolic two-temperature theory as well as in one-temperature theory (the Lord-Shulman model), thermal and mechanical waves spread at low speeds in the thermoelastic organization.

1. Introduction

The 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. Therefore, we will discuss some recent models that need to be addressed.

The thermoelasticity model has been developed by Chen and Gurtin, based on two different conductive and dynamic temperatures. The 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 [4–13]. 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]. Therefore, Youssef and El-Bary have changed this model and implemented a new two-temperature model based on different thermal conductivity rules, called hyperbolic two-temperature 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. The speed of thermal wave propagation is limited in this model. The 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.

The fact that fractional systems possess memory justifies this generalization, as the time evolution of romantic relationships is naturally impacted by memory [15, 16]. The 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]. Thus, 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 one-order. The intermediate spectrum of the heat gives different orders of thermoviscoelastic substances [19].

Depending on the factional-order strain, Youssef launched another new theory of general thermoelasticity. The relationship between stress and strain is seen as a new and distinct addition to Duhamel-Neumann [20]. Youssef has solved thermoelasticity with fractional sequence strain in one dimension, half of the space of Biot, Lord-Shulman, Green-Lindsay, and Green-Naghdi type II [20]. Youssef solved many applications of thermoelasticity of infinite thermoelastic spherical medium [21–24]. Mukhopadhyay and Kumar studied the general thermoelastic interactions with the cavity in the unlimited medium [25]. Many authors solved problems of thermoelastic medium with spherical cavity [26–35].

Most of the authors who regarded their applications as the spherical medium faced the singularity problem in its center. Few authors, for example, Thibault 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. The 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.

2. The Governing Equations

Consider a perfect, thermoelastic, conducting, and isotropic spherical body that occupies the region . We use a spherical coordinative system 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.

Figure 1 

The isotropic homogeneous thermoelastic solid sphere.

We note that, due to spherical symmetry, the displacement components have the form

The equation of motion [10, 20] is

The constitutive equations with damage mechanics variable [10, 20] are

The strain components arewhere is the cubical dilatation and it is given by

The hyperbolic two-temperature heat conduction equations take the forms [10, 14, 20]where is the hyperbolic two-temperature parameter [14] and .

The Riemann–Liouville fractional integral description is used in the above equations written in a convolution type form [20, 37]which provides Caputo with the form of fractional derivatives:

We consider that and are the conductive and dynamical temperature increments, respectively. Then equations (2)–(5), (9), and (10) take the forms

Equation (13) can be rewritten to be in the form

The following nondimensional variables are used for convenience [5, 9, 10]:

Then, we obtainwhere , , , , , , , and .

The primes are suppressed for simplicity.

The operator is singular at ; however, if symmetry prevails, the singularity will be reduced by the following L’Hopital’s rule [36]:

Then, we getsatisfying the boundary condition

Hence, we have

By applying the form in (26) in equations (19)–(21), we obtain

We apply the Laplace transform defined as

The Laplace transform of the fractional derivative is defined as [37]

The following initial conditions have been assumed:

Thus, equations (19)–(24) have the formswhere .

Substituting equation (35) into equations (33) and (34), we getwhere , , , and .

Substituting equation (41) into equation (40), we obtainwhere and .

3. 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 and .

Matrix A has four linearly independent eigenvectors; hence, we can construct a matrix V from the eigenvectors of matrix A such that , where is a diagonal matrix [38].

If we make the substitution in system (44), thenwhich giveswhere are the eigenvalues of matrix A or the roots of the characteristic equationwhere

Since W is a diagonal matrix, then system (46) is uncoupled, making each differential equation in the system have the form . The solution to each of these linear equations is . Hence, the general solution of system (46) can be written as the column vector [38]:

Then, 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

The boundary conditions in (27) and equation (52) give that

Hence, we obtain

To get the constants , we must apply the boundary conditions at ; we consider the sphere when is thermally shocked as follows:where is the Heaviside unit step function and is constant.

For the mechanical boundary condition, we consider the surface of the sphere when is connected to a rigid foundation, which can stop any displacement , offering zero volumetric deformation from equation (39); hence, we have

Applying Laplace transform on equations (56) and (57), we obtain

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) as

The singularity in (64) can be removed by using L’Hopital’s rule twice in a row as follows [36]: