1. Introduction
The thermoelastic responses of different structures with spherical cavities have received much attention because of their usefulness in many industrial applications. In the following, we restrict our attention to the application of continuums with spherical cavities. All the problems discussed are concerned with thermoelastic exchanges within the framework of several generalized thermoelasticity theories.
Generalized thermoelasticity models, with one or more relaxation times, have been proposed to modify the heat conduction equation. One of the original forms of the heat conduction equation, associated with gases theory, was introduced by Maxwell [
1]. Another form was proposed within the framework of heat conduction in rigid structures by Cattaneo [
2]. A third form was introduced by Dhaliwal and Sherief [
3] by extension to the case of an anisotropic medium. To overcome the contradiction of an endless velocity of thermal waves intrinsic to classical coupled thermoelasticity (CTE) theory [
4], attempts have been made by various investigators, for a range of reasons, to modify coupled thermoelasticity to entail a wave-type heat conduction equation.
Lord and Shulman (L–S) [
5] developed generalized thermoelasticity theory presenting one relaxation time in Fourier’s law of heat conduction equation and therefore converting it into a hyperbolic type. Banerjee and Roychoudhuri [
6] discussed the generalized theory of thermo-elasticity suggested by L–S [
5] to examine thermo-visco-elastic wave propagation in an unlimited viscoelastic body of Kelvin–Voight type with a spherical hole. Sinha and Elsibai [
7] discussed thermoelastic exchanges in an unlimited solid with a spherical inclusion considering L–S and G–L theories. Rakshit Kundu and Mukhopadhyay [
8] described field variables in a viscoelastic body with a spherical hole. Youssef [
9] described a problem of thermoelastic exchanges in a limitless body including a spherical hole subjected to a moving heat source according to L–S theory. Elhagary [
10] described the problem of a thermoelastic unbounded solid including a spherical hole in the framework of L–S diffusion theory. Karmakar et al. [
11] determined the temperatures, stress, displacement, and strain in an unbounded solid including a spherical hole in the framework of processes addressed by two-temperature theory (2TT).
Later, Green–Naghdi (G–N) [
12,
13,
14] created three versions for generalized thermoelasticity that were identified as I, II, and III types. Mukhopadhyay [
15,
16] presented thermoelastic exchanges in an unbounded solid including a spherical hole in the framework of G–N theory. Mukhopadhyay and Kumar [
17] considered thermoelastic exchanges in an infinite solid with a spherical hole in the framework of several theories. Allam et al. [
18] investigated electro-magneto-thermoelastic exchanges in an infinite solid with a spherical hole in the framework of G–N theory. Banik and Kanoria [
19] determined the thermoelastic quantities in an infinite solid with a spherical inclusion in the framework of the 2TT. Abbas [
20] investigated a general solution to the field equations of 2TT in an unbounded medium with a spherical hole in the framework of the G–N model. Bera et al. [
21] investigated the waves arising from the boundary of a spherical cavity in an infinite medium. Biswas [
22] examined the thermoelastic exchange in a limitless body including a spherical cavity in the context of the G–N model. Chandrasekharaiah and Narasimha Murthy [
23] considered thermoelastic exchanges in an infinite body including a spherical inclusion.
Green and Lindsay [
24] pioneered an additional theory, known as the G–L model, that included two relaxation times. Roy Choudhuri and Chatterjee [
25] studied spherically symmetric thermoelastic waves in an unbounded body containing a spherical hole. Sherief and Darwish [
26] presented a problem of a thermoelastic unbounded solid containing a spherical hole in the framework of thermoelasticity theory with two relaxation times. Mukhopadhyay [
27] discussed thermally induced vibrations of an unbounded viscoelastic body including a spherical hole in the framework of G–L theory. Ghosh and Kanoria [
28] determined thermoelastic quantities in a functionally graded (FG) spherically unbonded body including a spherical hole in the framework of G–L theory. Kanoria and Ghosh [
29] examined thermoelastic exchanges in an FG hollow sphere in the framework of the G–L model. Das and Lahiri [
30] considered a thermoelastic problem for an unbounded FG and temperature-dependent spherical inclusion in the framework of G–L theory.
Many investigators have used dual/triple-phase-lag (D/TPL) heat transfer theory to examine thermoelastic exchanges in unbounded mediums including spherical cavities. DPL theory was originally presented by Tzou [
31,
32] to describe some problems at a macroscopic scale. Abouelregal and Abo-Dahab [
33] presented thermal quantities in an unbounded solid with a spherical hole in the framework of DPL theory. Hobiny and Abbas [
34] applied DPL theory in the examination of photo-thermal exchanges in an infinite solid containing a spherical cavity. Mondal and Sur [
35] studied a coupled problem in an infinite solid with a spherical hole in the framework of a photothermal transport process in relation to 2TT. Singh and Sarkar [
36] examined thermoelastic exchange in a 2TT unbounded isotropic body containing a spherical cavity in the framework of a memory-dependent derivative (MDD). Comparisons were made graphically between the 2T TPL theory and 2T L–S theory with MDD. Many researchers have dealt with one-dimensional (1D) problems in generalized thermoelasticity in unbounded mediums with spherical cavities [
37,
38,
39,
40,
41,
42,
43,
44].
In the current article, magneto-thermoelastic exchanges in an infinite solid with a spherical hole are studied with respect to multi-time-derivative thermoelasticity theories [
45,
46,
47,
48,
49,
50,
51,
52,
53]. A refined DPL model is used for this purpose. The technique of Laplace transforms in the time domain is applied to obtain the governing equations analytically. The derived equations are solved and then Laplace inversion is carried out to obtain the field quantities numerically. For verification proposes, the outcomes are compared with those obtained previously. Additional results are presented graphically and others are reported for future comparison.
2. Basic Equations
Let us be concerned with thermoelastic analysis of an isotropic body including a spherical cavity of radius
R based on unified multi-phase-lag theory. It is assumed that the outer edge of the spherical cavity is traction-free and subjected to harmonically varying heat (See
Figure 1). The spherical cavity coordinate system (
r,
θ,
ϕ) is used to address the present problem.
Figure 1.
A spherical cavity in an unbounded medium under harmonically varying heat and external magnetic field.
Figure 1.
A spherical cavity in an unbounded medium under harmonically varying heat and external magnetic field.
The governing equations for a linear isotropic homogeneous thermoelastic body in the absence of volume forces are given by:
where
and
are the stresses and strains and
denotes Kronecker’s delta tensor.
is considered in the context of the refined thermoelasticity form in which
and
denote the following higher-order time-derivative operators:
Equation (3) with the aid of Equation (2) are the more general ones when has numerous integers more than zero. Some specific cases may be achieved as
- (i)
Dynamical coupled thermoelasticity (CTE) model [
4]:
and
,
- (ii)
Lord and Shulman (L–S) model [
5]:
,
and
,
- (iii)
Green and Naghdi (G–N) model without energy dissipation [
12,
13,
14]:
,
,
,
and
,
- (iv)
The simple dual-phase-lag (SDPL) model [
50,
51,
52]:
,
and
,
- (v)
The refined with dual-phase-lag (RDPL) model [
50,
51,
52]:
,
, and
,
The displacements of the present, axially symmetric spherical medium are summarized as
The non-vanishing strains and volumetric strain can be expressed as
Thus, the volumetric strain
has the form
The constitutive equations for the spherical symmetric system can be stated as
Applying the operator (
) to both sides of Equation (15), one gets
in which
denotes the Laplacian operator in spherical coordinates. It meets the formulation
4. Closed-Form Solution
The comprehensive solutions are provided by resolving Equations (21) and (22) to obtain, firstly, temperature
and volumetric strain (dilatation)
. Then, the subsequent radial displacement and thermal stresses may be presented as functions of
and
. For this objective, we will first employ the next initial conditions:
In adding together to the above homogenous initial conditions, we also used the thermomechanical boundary conditions. The current unbounded body will be studied as quiescent and the surface of the spherical cavity is assumed to be exposed to constant heat and traction free. Such conditions can be explained as
Moreover, we take into consideration the following regularity conditions
The Laplace transform is carried out for Equations (19)–(22), and, with the homogeneous initial conditions that appeared in Equation (24), one gets:
where
The system of equations provided in Equations (30) and (31) can be indicated in the differential equation
where the coefficients
are given by
and the temperature
is reformed as follows
Equation (33) is very complicated since it is presented in a polar coordinate system. It can be expressed as
where
are the roots of
These roots
are given, respectively, by
Equation (36) tends to the next modified Bessel’s equation of zero-order
which has a solution under the regularity conditions:
,
as
. Therefore, the general solution of Equations (35) and (39), that is bounded at infinity, is provided by
where
are integration parameters and
Using the relation between
and
one can pick up the solution for the dimensionless form of radial displacement pretending that
disappears at infinity as:
where
Up to here, the solution is finished. It is as much as needed to establish the two parameters
with the aid of the boundary conditions given in Equations (25) and (26). So, one gets
Therefore, the current analytical solution is already provided for the modified formulations in Laplace space. To achieve the solution in the basic time-space one can consider a function
as an inversion of the Laplace function
in the form
where
is an arbitrary constant,
is the real part,
suggests the imagined number unit and
denotes a sufficiently big integer. For faster combination, various numerical analyses have shown that the approximation of
fulfills the connection
[
35]. The numerical procedure cited is used to invert the terms of temperature
, radial displacement
, volumetric strain
, radial stress
, and circumferential stress
.
6. Conclusions
The present refined dual-phase-lag model is innovative and produces accurate results for variables such as temperature, volumetric strain, displacement, and stresses. The multi-time derivatives heat equation was explained. The constitutive relations of spherical coordinates were considered to examine the thermoelastic coupling behavior of an infinite medium with a spherical cavity due to a uniform heat. To create a unified model, one can combine other models, including the coupled dynamical thermoelasticity model, the Lord and Shulman model, the Green and Naghdi model without energy dissipation, as well as a simple dual-phase-lag model. The system of two high-time-derivative differential coupled equations was solved, and all field variables were developed for the thermoelastic coupling response of an infinite medium with a spherical hole. Various confirmation examples and applications were offered to compare the outcomes due to all models with the refined ones. A sample set of graphs were presented to demonstrate relationships of variables through radial direction of a spherical hole. Some tables have been provided as confirmation examples to provide benchmark outcomes for future comparisons by other researchers. The described and demonstrated outcomes revealed different behaviors of all field variables and dimensionless time parameters. The present dual-phase-lag theory diminished the magnitudes of the examined variables, which may be significant in some practical applications. The G–N model provided appropriate outcomes over a small range. However, the refined model produced improved and exact outcomes.